This book provides the first systematic treatment of the thermodynamic theory of site-specific effects in biological macromolecules. It describes the phenomenological and conceptual bases required to allow a mechanistic understanding of these effects from analysis of experimental data. The thermodynamic theory also results in novel experimental strategies that enable the derivation of information on local, site-specific properties of a macromolecular system from analysis of perturbed global properties. The treatment focuses on binding phenomena, but is amenable to extension both conceptually and formally to the analysis of other cooperative processes, such as protein folding and helix-coil transitions. Much attention is devoted to the analysis of mutational perturbations used as probes of site-specific energetics. The arrangement of various sections in the book allows its use both as a reference work on cooperative binding phenomena for any scientist involved in structure-function studies of biological macromolecules, and as a text for graduate students in biochemistry and biophysics.
THERMODYNAMIC THEORY OF SITE-SPECIFIC BINDING PROCESSES IN BIOLOGICAL MACROMOLECULES
THERMODYNAMIC THEORY OF SITE-SPECIFIC BINDING PROCESSES IN BIOLOGICAL MACROMOLECULES ENRICO DI CERA Washington University School of Medicine
CAMBRIDGE UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Cambridge University Press 1995 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1995 First paperback edition 2005 A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data Di Cera, Enrico. Thermodynamic theory of site-specific binding processes in biological macromolecules / Enrico Di Cera. p. cm. Includes bibliographical references. ISBN 0 521 41659 0 (hardback) 1. Binding sites (Biochemistry)—Thermodynamics. 2. Ligand binding (Biochemistry)-Thermodynamics. 3. Cooperative binding (Biochemistry)-Thermodynamics. 4. Macromolecules-Thermodynamics. I. Title. QP517.B42D5 1995 574.192-dc20 95-15257 CIP ISBN 0 521 41659 0 hardback ISBN 0 521 61975 0 paperback
It is certainly not the least charm of a theory that it is refutable; it is with precisely this charm that it entices subtler minds. Nietzsche, Beyond Good and Evil
To Antonella and Leonardo
Contents
Preface
xiii
1 Statistical thermodynamic foundations 1.1 Postulates and basic ensembles 1.2 The generalized ensemble 1.3 The laws of thermodynamics 1.4 Fluctuations 1.5 Simple models for energy transition 1.6 Simple models for binding
1 1 18 26 34 37 41
2 Global binding processes 2.1 The reference system 2.2 Cooperative binding 2.3 Properties of the binding curve 2.4 Properties of the binding capacity 2.5 Linkage effects: The reference cycle 2.6 Linkage effects: General treatment involving two ligands
47 47 57 68 76 88 99
3 Local binding processes 3.1 The reference cycle and site-specific cooperativity 3.2 Contracted partition functions 3.3 Thermodynamic basis of site-specific effects 3.4 Properties of the binding curve in the local description 3.5 Properties of the binding capacity in the local description
108 108 123 139 146 152
4 Specific cases and applications 4.1 The case N = 2 4.2 The case N = 3
158 158 169 XI
xii
Contents
4.3 The case N = 4 4.4 Site-specific properties of hemoglobin 4.5 Site-specific binding curves from global properties of structurally perturbed systems 4.6 Pairwise coupling
182 190 214 230
5 Site-specific effects in Ising networks 5.1 Transition modes 5.2 A combinatorial model for molecular recognition 5.3 Site-specific effects in Ising networks 5.4 The probe theorem
245 246 258 265 279
References
285
Index
292
Preface
Biological function in a macromolecular system often arises from the contribution of many constituent structural domains that communicate in a cooperative fashion. Formally, a global macromolecular property, F, can be cast as follows where the /s depict the contribution of individual structural domains that behave as subsystems open to interaction with the rest of the system. Relevant examples of global properties are: protein stability, where the /s represent the contribution of particular folding units to the macroscopic free energy of unfolding; helix-coil transitions, where the /s represent the helicity of individual residues and their contribution to the helix state of the peptide as a whole; binding and linkage phenomena, where the /s denote the probability of binding to individual sites and F is the average number of ligated sites. In all these cases, it is the cooperative behavior of the individual constituents that sets the behavior of the system as a whole. Also, the code for cooper ativity and structure-function relationships is embodied by the local properties / s , rather than F. Consequently, a description of cooperative phenomena must be sought for in terms of local rather than global quantities. The limitations of a global description are demonstrated by the rather obvious fact that F is uniquely defined once the local properties /s are known, but the reverse is not true in general. Hence, the detailed aspects of the communication among constituent domains of a macromolecule that encapsulate the connection between structure and function are often swamped in the global picture and crucial information on mechanisms of cooperativity is lost. The need for a description of cooperative effects in terms of local properties has been recognized for a long time, since the pioneering Xlll
xiv
Preface
studies of Wegscheider (1895) on the ionization reactions of polybasic substances. For many years, however, our understanding of cooperative phenomena has remained limited to effects arising in the global picture due to the limitations imposed by experimental techniques. Likewise, theoretical treatments of cooperative effects in biology have so far been focused on the analysis of global effects. Recent advances in various areas, and especially in X-ray crystallography, NMR spectroscopy and recombinant DNA technology, have made it possible to access information at the local level and introduce a wide range of site-specific perturbations of macromolecular systems in a way never before possible. Crucial events pertaining to global phenomena can now be dissected in terms of the contribution of individual binding sites, folding units, amino acid residues or even atoms, thereby revealing the true and extraordinary complexity of cooperative effects in biology. These exciting new developments reinforce the notion that a predictive understanding of function and energetics from structure can only be gained from information on the local properties of a biological macromolecule and the network of communication among its constituent domains. A theoretical treatment of cooperative effects arising at the local, site-specific level is therefore both timely and important. This monograph provides the first systematic treatment of the thermodynamic theory of site-specific effects in biological macromolecules. It describes the phenomenological and conceptual basis to gain a mechanistic understanding of these effects from analysis of experimental data. The theory also brings about novel experimental strategies of deriving information on local properties of a macromolecular system from analysis of perturbed global properties. Although the treatment focuses on binding phenomena, it is amenable of extension both conceptually and formally to the analysis of other cooperative processes, like protein folding and helix-coil transitions. Much attention is also devoted to the analysis of mutational perturbations, used as probes of site-specific energetics. Chapter 1 gives an introduction to basic thermodynamic concepts that form the backbone of the phenomenological theory of ligand binding. Cooperativity and linkage in the global description are dealt with in Chapter 2. Most of this chapter is a summary of the classical treatment of ligand binding and linkage effects detailed in excellent monographs (Hill, 1984; Wyman and Gill, 1990). The treatment of local, site-specific effects is introduced in Chapter 3 as an extension of concepts introduced in Chapters 1 and 2. The formalism and conceptual framework based on contracted partition functions are also introduced in Chapter 3. Applications of the theory of
Preface
xv
site-specific effects are given in Chapter 4. The case of hemoglobin cooperativity is discussed in Section 4.4. A novel strategy of approach to site-specific effects based on the analysis of global properties of structurally perturbed systems is introduced in Section 4.5. A general approach to site-specific cooperativity based on the analysis of pairwise coupling patterns derived from mutational perturbations is given in Section 4.6. Finally, Chapter 5 addresses the rather difficult problem of site-specific effects in Ising networks. These networks can be used as models for molecular recognition and serve to explore rules for site-site communication in extended systems that cannot be analyzed within the framework of the phenomenological theory. Section 5.4 describes possible new approaches to the study of the Ising problem in two and higher dimensions using site-specific thermodynamics. The arrangement of various sections of this monograph allows for a great deal of flexibility in using it as a reference for cooperative binding phenomena, or as a textbook for graduate students in biochemistry and biophysics curricula. The reader thoroughly familiar with ensemble theory and the classical Hill-Wyman treatment of binding and linkage can start with Chapter 3, although reading of Section 1.2 and Section 2.4 is recommended. The reader interested mostly in applications of the theory to binding phenomena and mutational perturbations can start with Sections 4.4-4.6 and then move to Sections 5.1 and 5.2, although reading of the basic Sections 3.1 and 3.2 is also recommended. In using the monograph as a textbook in biochemistry and biophysics curricula, the following 'pathways' are recommended: Biochemistry Chapter Chapter Chapter Chapter Chapter
1 2 3 4 5
Sections 2.1-2.3, 2.5 Sections 3.1,3.2 Sections 4.2, 4.4-4.6
Biophysics Sections 1.5, 1.6 Sections 2.1-2.3, 2.5, 2.6 Sections 3.1-3.3 Sections 4.2, 4.4-4.6 Sections 5.3,5.4
In either pathway the monograph will serve its purpose of providing the conceptual tools for deciphering cooperativity codes in biological macromolecules. These tools should be acquired in conjunction with the recent developments in spectroscopic techniques and recombinant DNA technologies by those students who seek to obtain a more integrated training in biochemistry and biophysics. Knowledge of calculus and elementary differential equations is required in either pathway. The requirement can be reduced to a minimum in a short course emphasizing the practical applica-
xvi
Preface
tions of site-specific thermodynamics, which may include Sections 3.1, 3.2, 4.4-4.6 only. Such a module may be integrated in the core course of a biochemistry curriculum. A more demanding course emphasizing the statistical thermodynamic foundations can be prepared by addition of Sections 1.1-1.4 to the biophysics pathway. Much of the inspiration to develop the theory dealt with in this monograph has come from the landmark experimental work of Gary Ackers on hemoglobin (Ackers, Doyle, Myers and Daugherty, 1992) that has led to the first, most accurate and complete dissection of the sitespecific properties of a cooperative system in biology. The author is greatly indebted to Professors Gary Ackers and Timothy Lohman for reading parts of the monograph and providing helpful suggestions, and to Professor Michele Perrella for providing experimental data prior to publication and helpful comments on Section 4.4. The author is also grateful to Professors Ken Dill, Neville Kallenbach, Eaton Lattman and George Rose for helpful discussions on some of the topics covered in the monograph. A very special thanks goes to Professors John Edsall and Bruno Zimm for several comments that were invaluable in the final preparation. Professor Jeffries Wyman and the late Professor Stanley Gill provided excellent advice in the early stages of preparation of the monograph. Helpful comments also came from many graduate students and postdoctoral fellows here at Washington University and especially from Quoc Dang, Karl-Peter Hopfner, Sean Keating, Yong Kong and Murad Nayal while in the author's laboratory. The skillful secretarial assistance of Anna Goffinet was invaluable throughout the preparation of the monograph and Sophie Silverman carefully proofread various versions of the manuscript. Finally, the author gratefully acknowledges partial support from National Science Foundation Grants DMB91-04963 and MCB94-06103 during the period in which the monograph was written. St. Louis,
E. Di Cera
1 Statistical thermodynamic foundations
We are interested in a system composed of a biological macromolecule and a number of ligands, of which the solvent can be considered one, under conditions of temperature and pressure of biological relevance. Our goal is to characterize the behavior of the macromolecule in its interaction with the various components of the system and the rules underlying the mutual interference of physical and chemical variables. In this chapter we deal with the statistical thermodynamic foundations of binding processes and the concepts that form the basis of our treatment. 1.1 Postulates and basic ensembles
The physico-chemical properties of a system are defined thermodynamically by a set of macroscopic quantities accessible to experimental measurements (Fermi, 1936; Schrodinger, 1946). If the macromolecule is taken as the system, then all observables reflect properties of the system and the way it is affected by physical and chemical driving forces. A macromolecule in the presence of multiple ligands can be seen as a system existing in a number of distinct energy states, Eu E2, . . . Er, totally analogous to the energy states of a quantum mechanical system. Each energy state may be coupled to a conformational state of the macromolecule, i.e., a specific arrangement of its secondary, tertiary or quaternary structure. In addition, Ej is specified by the number of ligands bound, or by a particular ligated configuration. Different energy states may group together in the same energy level according to their degeneracy. We consider a prototypic system composed of the macromolecule and the solvent, in contact with a heat bath at constant temperature 7\ and focus our attention on a particular configuration of the macromolecule (our system) with TV water molecules (the solvent) bound to it, as shown in 1
2
1 Statistical thermodynamic foundations
Figure 1.1. We use the term bound in a thermodynamic sense, to distinguish the water molecules in any form of interaction with the system from those belonging to the bulk solvent. In what follows, we make no attempt to discriminate among the various binding modes that characterize protein hydration, whether specific or non-specific. We use the macromolecule in solution to illustrate the salient thermodynamic features of the simplest system of interest. If V is the volume of the macromolecule, then any energy state can be written as Ej(N, V) and the spectrum of energy values Eu E2, . . . Er denotes the possible alternative states accessible to
Figure 1.1 Connolly surface of lysozyme, constructed from the crystal structure (Weaver and Matthews, 1987). Water molecules are indicated by solid dots. Some molecules are present as components of the bulk solvent. Other molecules are 'bound' to the accessible surface of the enzyme and make polar contacts with specific residues.
1.1 Postulates and basic ensembles
3
the macromolecule with specified values of V and N. The question that arises at this point concerns the probability of finding the macromolecule in a particular energy state, £,, when the temperature of the heat bath is held constant. To solve this problem we use the ensemble method introduced by Gibbs (Gibbs, 1902, 1928; Hill, 1960). Consider an ensemble of a very large number of identical replicas of our system, as depicted in Figure 1.2, with the same value of V and N, and in thermal contact with each other. Each system is in contact with a heat bath at constant T and is allowed to exchange heat, but not water molecules, with the surrounding systems. After equilibrium is reached and T is uniform everywhere, the entire ensemble of systems, or supersystem, is thermally insulated by an adiabatic wall. If F is the total number of replicas in the supersystem, then each system can be thought of as being in contact with a heat bath at temperature T formed by the remaining F — 1 systems. The supersystem artificially generated by our replication process is an isolated system that cannot exchange heat or matter with the environment, as opposed to each constituent system of the ensemble that is closed and isothermal, since it is allowed to exchange heat, but not matter, with the environment. The properties of our original system can be derived from the properties of the supersystem characterized by a
Figure 1.2 Schematic representation of a canonical ensemble. A number of elementary systems with fixed values of V and N are in thermal contact with each other at constant temperature. The supersystem formed by F elementary systems is insulated by an adiabatic wall from the environment.
4
1 Statistical thermodynamic foundations
volume TV, TN water molecules and a total energy Et. In the thermodynamic limit F —> °°, the supersystem is composed of an extremely large number of replicas and the properties of each system separately can be derived as 'ensemble averages' taken over the entire collection of replicas. The validity of this assertion is supported by two postulates of Gibbs. The first postulate states that: The long time average of a variable in the system at equilibrium is equal to its ensemble average in the thermodynamic limit F -» oo 9 provided the ensemble of systems replicates the system of interest and its environment. This postulate is intuitively obvious. It simply states the equivalence between averages obtained by monitoring the behavior of the system at equilibrium over a long time scale and those obtained over the ensemble artificially constructed. The time evolution of a variable of interest, say the energy of the system, is plotted in Figure 1.3 as a trajectory whose
CO
d
CO
d
d
o d 0.0
20.0
40.0
60.0
80.0
100.0
Time Figure 1.3 Time evolution of the energy (in arbitrary units) of a canonical ensemble at equilibrium. The time average of the energy values gives the thermodynamic measure of the energy E.
1.1 Postulates and basic ensembles
5
time average gives the thermodynamic measure of E{N, V). The contact with the heat bath makes the energy fluctuate and assume any of the allowable values in the spectrum. If we were to monitor the energy for our system at equilibrium, we would obtain a plot similar to that shown in Figure 1.3. The average value of the energy must, of course, be independent of the particular time at which the observation is started. Hence, we could make repeated observations about our system and obtain a bundle of trajectories such as the one shown in Figure 1.3, each of which would yield the same value of the average energy E(N, V). We may think of each trajectory as belonging to a replica of our original system. In the limit where this number becomes arbitrarily large, we can take a snapshot of the bundle of trajectories at any given time and display the energy values for all individual systems, as shown in Figure 1.4. The average of these
CO
d
CO
d
s? 88
CN
d
q d 0.0
20.0
40.0
60.0
80.0
100.0
System Figure 1.4 Spectrum of energy values for a bundle of trajectories such as the one shown in Figure 1.3, taken at a fixed time, for a canonical ensemble at equilibrium. The abscissa enumerates a representative sample of systems in the supersystem in Figure 1.2. The ensemble average of the energy values is equivalent to the time average derived from Figure 1.3 and gives the thermodynamic measure of the energy E.
6
1 Statistical thermodynamic foundations
values must be independent of the particular time chosen. Sections of the bundle taken at different times must yield energy profiles with the same average value, and this value is evidently the same as that obtained in the long time average of a single trajectory. Hence the first postulate, which establishes a complete equivalence between time averages over the system and ensemble averages independent of time over a supersystem constructed of a large number of replicas of the original system. In order to determine the properties of our system specifically we need a second postulate, also known as the 'principle of equal a priori probabilities' for an isolated system, that states: In an isolated system for which N, V and the energy E are specified, all energy states have equal a priori probabilities. All allowable energy states, consistent with the specified values of N 9 V and E, occur with the same frequency if the behaviour of the system is followed over time or, equivalently, the system spends equal amounts of time in any of the allowable energy states. This important consequence of the two postulates is known as the 'ergodic hypothesis' and plays a central conceptual role in statistical mechanics and its mathematical foundations (Tolman, 1938; Khinchin, 1949; Kubo, 1965; de Groot and Mazur, 1984). In an ensemble representative of an isolated system, the probability of occurrence of any energy state is exactly the same for all systems and if any system is chosen at random, it will be in any of the allowable energy states with equal probability. The second postulate as stated above applies only to isolated systems, while the first postulate holds quite generally for any equilibrium system. We are now in the position to find the properties of our original system composed of the macromolecule with a volume V, a number N of water molecules bound, and in contact with a heat bath at constant temperature T. The ensemble of elementary systems shown in Figure 1.2 is, by construction, an isolated system which must obey the Gibbs postulates. The energy of the supersystem is njEj
(1.1)
7= 1
where n; is the number of systems in the energy state Ej. Also,
r = i>,y=i
(1.2)
1.1 Postulates and basic ensembles
7
is an obvious conservation relationship. There is a very large number of distributions, Z), of rij values consistent with eqs (1.1) and (1.2), and specifically there are as many as Q(D) =
(1.3) . . . nr(D)\ ni{D)\n2{D)\ possible ways of realizing the particular distribution, D, of such values. Here ny(D) denotes the number of systems in state Ej for the particular distributions D. The probability, ^ 7 , of finding the macromolecule in a given energy state Ej is the weighted mean of all possible rij(D) values, divided by the total number of systems, i.e.,
(1-4) The distribution of rij values, Q(D), is multinomial and goes into a Gaussian for large T by virtue of the De Moivre-Laplace theorem (Wilson, 1911; Feller, 1950). In the limit r~* °o? the Gaussian goes into a completely sharp Dirac <5-function centered about the most probable distribution D = D*. Since the 5-function vanishes everywhere except for D = D*, then the simple result t//;- = nf/T is obtained from eq (1.4), where n*= «,(£>*) is the value of n } in the most probable distribution. The value of n*can be found by maximizing Q, or equivalently lnQ, subject to the constraints in eqs (1.1) and (1.2). Application of the method of Lagrange's undetermined multipliers (Wilson, 1911) gives —[lnQ(D) -
ocY - l3Et] = 0
(1.5)
where a and (5 are the undetermined multipliers. Using the Stirling approximation lnx! — x\nx — x for the factorial terms of Q(D) in eq (1.3), and recalling that T and Et are functions of nj9 yields lnT - \nnf- a - /3Ej = 0 (1.6) Hence, V/,. = -^*=exp(-(*-/SE y )
(1.7)
gives the probability that the macromolecule exists in the energy state Ej defined by V and N. The value of (3 is ( ^ r ) " 1 , where kB is the Boltzmann constant (Tolman, 1938; Hill, 1960). The value of a is
8
1 Statistical thermodynamic
foundations
obtained from the conservation relationship (1.2), which embodies the obvious fact that the sum of all ipj must equal unity, so that (1.8) Hence,
ft =
-.
^
\
(1-9)
gives the explicit expression for the probability of the macromolecule existing in the energy state Ej. This quantity decreases exponentially with the value of Ej. The ensemble average over all possible energy states ^
(1.10)
7= 1
defines the energy of the system in the thermodynamic sense. The closed isothermal system considered in the foregoing analysis is called a canonical ensemble and the probability distribution in eq (1.9) is the Boltzmann distribution for a canonical ensemble (Tolman, 1938). The properties of the macromolecule considered as a closed isothermal system, or a canonical ensemble, are completely specified by its volume V, the number of water molecules bound N and the temperature of the heat bath T. Thermodynamically, such a system is an N-V-T ensemble. We now introduce an important quantity that allows us to compute all thermodynamic functions of interest for the system. This is the canonical partition function defined as the sum
Z{N, V,T) = i exp [- Ej[N' V) ]
(1.11)
L k BT J ;= i The partition function enumerates all possible energy states of the system as they appear in the Boltzmann distribution. The relative contribution of each term is weighted exponentially by the Boltzmann factor associated with it. The probability of any energy state, Ej, is given by the ratio between the term containing the value of Ej and the partition function, as seen in eq (1.9). Mathematically, this can be expressed in compact form with the use of the partial derivative
1.1 Postulates and basic ensembles ,81nZ BEt
V; = ~
BF dEj
(1.12)
where F(N, V, T) = -kBTlnZ(N,
V, T)
(1.13)
is the Helmholtz free energy of the macromolecule and plays the role of the potential associated with the partition function Z. The energy of the macromolecule can be derived in an analogous way as
'<£ E =
\
BT JN,V
=-r-
(1.14)
BT N,V
A,
N,V
\ Tlto derive the Since Z is also a function of V and N, it is important quantities associated with these independent variables. The derivative BV
}N,T
\BVJN,T
(1.15)
gives the pressure of the system and '3JnZ_\ (BF\ (116) dN JV,T \ dN )V,T is the chemical potential of water. The change of Z or F with respect to the external conditions subject to experimental control yields information on the quantities associated with them. The energy of the system is obtained as the 'response function' to a change in temperature. Likewise, the pressure and chemical potential of water are obtained as responses to changes of V and N respectively. From the definition of F in eq (1.13) it also follows that (3F\ F- E = _k ln z _ r/31nZ\ = = -S (1.17) B \dT/N,v \ dT )N,V T The entropy, 5, of the macromolecule is derived from the change of the Helmholtz free energy as a function of temperature. An explicit expression for S is obtained from the definition of t/>; as follows. Taking the logarithm of t/>/ one has In y. = —^L - In Z (1.18) kBT Multiplying both members by xpj and summing over all values of j leads to = ^ipjlntyj ^B1
7= 1
(1.19)
10
1 Statistical thermodynamic
foundations
Hence,
provides an important definition of the entropy in terms of the probability of occurrence of the energy states, and the thermodynamic potentials E and F. Eqs (1.15)—(1.17) also allow for a definition of the thermodynamic potential F in differential, or Pfaffian form (de Heer, 1986) as follows
M() v,T
\dV/N,T
4 + M \dT/N,V
= ti&N - PdV - SdT (1.21)
This Pfaffian form summarizes the properties of the macromolecule as a canonical ensemble. The integral form of F is F= E - TS
(1.22)
and follows directly from eq (1.20). Having defined the properties of our original closed isothermal system, we turn to the supersystem itself which is an isolated system for which N, V and the total energy E are fixed. Consider all systems in the supersystem that belong to the same energy state £";, group and surround them by an adiabatic wall. This yields an ensemble of systems with the same value of N9 V and E = Ej. Such an ensemble is called microcanonical and differs from the canonical ensemble insofar as all systems in the ensemble have the same energy. Application of the same arguments developed for the canonical ensemble leads to the conclusion that \pj must be independent of j in the microcanonical ensemble. This is because the microcanonical ensemble is a degenerate canonical ensemble where all systems have the same energy E = E1 = E2 = . . . = Er. We also know, by construction, that the microcanonical ensemble is a subensemble of the canonical ensemble in Figure 1.2, where all systems with the same energy have been grouped together and isolated. Hence, the degeneracy Q(D*) of the most probable distribution D* in a microcanonical ensemble with energy E = Ej coincides with the value of it* in the canonical ensemble. The value of the probability ip for the energy E is simply l/Q(D*) and is the same for all systems of the ensemble since they belong to the same energy level and there are a total of Q(£>*) such systems. This follows directly from the Boltzmann distribution in eq (1.9) by letting Ej = E for all j = 1, 2, . . . Q(D*). The partition function of the macromolecule as a microcanonical ensemble is given by Q and is completely defined once N, V
1.1 Postulates and basic ensembles
11
and E are fixed. The thermodynamic potential associated with Q is the entropy 5 and is obtained from eq (1.20) by letting ^? = tp = 1/Q for all 7 = 1,2, . . . Q , i.e., (1.23) S(N, V, E) = kB\nQ(N, V, E) The microcanonical ensemble is therefore an N-V-E ensemble. The Pfaffian form associated with the potential 5 is 3N JV.E
\dVjs,E
\BE}N,V
= —^dN + £-dV + — &E (1.24) T T T and is a consequence of eq (1.22). In fact, taking the differential of eq (1.22) yields dF = dE - TdS - SdT = -SdT - PdV + pdN (1.25) and hence eq (1.24). The response functions of the macromolecule treated as a microcanonical ensemble are derived from eq (1.24) as follows
) BE
sv
JN,V
)
(
\BEJN,V
(1.27)
)
jv,E
f)
\dN)v,E
(
They provide an alternative way of characterizing the pressure and chemical potential of water using the response of the macromolecule to variables controlled externally. The Pfaffian form of the energy of the macromolecule is implicit in the differential form of the entropy in eq (1.24) and is given by d£ = TdS - PdV + tidN
(1.29)
This form will be very useful in the following sections. Note that the energy, like the entropy, is a function of extensive quantities only and therefore scales linearly with the dimensions of the system. We will analyze the important consequences of this fact in Section 1.3. The microcanonical and canonical ensembles provide two important conceptual frameworks for characterizing the properties of the macromolecule in our original system. When we look at the macromolecule as a microcanonical ensemble, we focus our attention on the properties of a
12
1 Statistical thermodynamic
foundations
configuration of volume V, with N water molecules bound and a particular energy state E, which is fixed. On the other hand, the macromolecule as a canonical ensemble is a system of fixed volume V, and N water molecules bound, but an energy E that can assume any value out of a spectrum of discrete energy states compatible with the preassigned values of N and V. The canonical partition function (1.11) encapsulates all the energy states of the system. If multiple states belong to the same energy level Ej, then Ej is said to be degenerate and Q(N, V, Ej) is the number of configurations belonging to E^ for N and V preassigned. The canonical partition function can be cast in two equivalent forms, depending on whether the summation is carried out over energy states or energy levels with their appropriate degeneracy taken into account, i.e.,
V, T) - iL> pkT[ - ^ > l - tow V, (1.30) The partition function cast in terms of energy levels shows the contribution of microcanonical components to the overall sum, as embodied by the microcanonical partition function Q. This term reflects the properties of a macromolecule of fixed volume V, with N water molecules bound and belonging to a fixed energy level Ej. The canonical partition function is the sum of several microcanonical partition functions weighted according to a Boltzmann factor. This is consistent with the fact that a canonical ensemble is merely a collection of microcanonical ensembles. The partition function enumerates all possible configurations consistent with the constraints imposed upon the system. In the case of the microcanonical ensemble we deal with the degeneracy of the energy level E assigned to the macromolecule of the volume V and with N water molecules bound. In the case of the canonical ensemble, we deal with the entire spectrum of energy values allowed by preassigned values of V and N. Analogous considerations can be applied to the case where different possible values of N are considered. In this case the macromolecule is studied as a system for which V is fixed, but the number of water molecules N and the energy E can both change. Since the energy level Ej(N, V) depends on N and V, it is expected to vary when N changes from zero to its upper bound M. Likewise, the entire energy spectrum is expected to change with N. In this case we seek to define the average value of the energy E in the system, as well as the average number of water molecules bound to the macromolecule, (N). Intuitively, we expect the partition function of the macromolecule to be the sum of a number of
1.1 Postulates and basic ensembles
13
canonical partition functions over the discrete variable N, weighted by a proper term that needs to be determined. This is by analogy with the transition between the microcanonical and canonical ensembles, where the Boltzmann factors containing the energy terms E}- were introduced as weighting factors of the microcanonical contributions to the canonical partition function. Consider the supersystem in Figure 1.5 constructed as in the case of the canonical ensemble by grouping together a very large number of replicas of our system of interest. Notice that the wall between adjacent systems has been modified to allow systems to exchange not only heat, but also matter in the form of water molecules. The system of interest is now open and isothermal and defines a grand canonical ensemble. The heat bath sets the temperature T of the system at equilibrium and a reservoir of water molecules (the solvent) sets the chemical potential of water \i. We allow a long time for equilibrium of the supersystem and then compute the most probable distribution of energy and N values for our system as in the case of the canonical ensemble. Likewise, we maximize lnQ[D(N)] using the method of undetermined multipliers. Here D(N) is the most probable
r
r
r
r
r
r
r
r
r j
V
Figure 1.5 Schematic representation of a grand canonical ensemble. A number of elementary systems with fixed values of V are in thermal contact with each other at constant temperature. Each system can also exchange molecules with its neighbors. The supersystem formed by F elementary systems is insulated by an adiabatic wall from the environment.
14
1 Statistical thermodynamic foundations
distribution of systems with energy states EX{N, V), E2(N, V), . . . Er(N, V) when N is assigned, and we have to consider all possible values of N. The constraints to be taken into account are evidently M
r
N=0j=l
N=0j=l
^
(1.33)
N=Oj=l
where rij(N) is the number of systems in the energy state Ej(N, V) and Nt is the total number of water molecules in the supersystem. This is the only additional constraint in the grand canonical ensemble compared to the canonical ensemble. A condition analogous to eq (1.5) states that [lnQ[D(A0] - aT- pEt - yNt] = 0
(1.34)
with Q[D(N)] =
(1.35) [D(N)]\n [D(N)]\ . . . n [D(N)]\ ni 2 r Here itj[D(N)] indicates the number of systems with N water molecules bound and in the energy state Ej(N, V), in the particular distribution D(N). Solution of eq (1.34) is straightforward and leads to In T - In nj(N)* - a - /3Ej(N, V) - yN = 0 (1.36) where nj(N)* is the most probable value of nj(N). The probability of finding the macromolecule with N water molecules bound and in the energy state Ej(N, V) is =exp \-a+ —
r
^—-\
L
kBr
(1.37)
\
The undetermined multipliers /3 and y have been expressed in terms of the appropriate driving forces as p=l/kBT and y = —[i/k BT. Again, the value of a is determined from the normalization condition for the probabilities ips as
i N=o/=i Hence,
i
y L
0.3(0
] k BT
J
1.1 Postulates and basic ensembles
15
[ Nfi — E;(N, V) ]
exp
I
ir^T
I (1.39)
The energy and the number of water molecules bound to the macromolecule in the grand canonical ensemble are fluctuating quantities and can only be defined as ensemble averages once the definition of ipj(N) is available. The relevant expressions are M
r
E= ^n>i(N)Ej(N,V)
(1.40)
N=0j=l
w
M
r
(L41)
= SSOT^ ;v=o/=i
The double sum in the logarithm of (1.38) and in the denominator of (1.39) is the grand canonical partition function
^ N=OJ=I
y L
(1.42) kT
J
The grand canonical partition function enumerates all possible states with respect to the number of water molecules bound and the allowable energy states. Two alternative forms of eq (1.42) deserve particular attention. Upon rewriting S in terms of energy levels and their degeneracy as follows
E(V, T, M) = 2 2Q(N> N=OJ=I
V
>^ e x p f ^ - ^ ^ l L
kBT
J
(1.43)
the contribution of microcanonical and canonical components becomes clear. Starting with a microcanonical ensemble for which N, V and Ej are fixed, the partition function Q is nested in a series expansion containing all possible values of Ej subject to the fixed values of V and N. Each term of this series, defining the canonical partition function, is weighted according to the Boltzmann factor. Next, the canonical partition function is nested in another series expansion over all possible values of N to generate the grand canonical partition function. The alternative form for the grand canonical partition function is a polynomial expansion in the water activity M
E(V, T, w) = X Z{N, V,T)wN N=0
(1.44)
1 Statistical thermodynamic foundations
16
The coefficients of this expansion are merely the canonical partition functions for a given set of values of V, N and T. The thermodynamic potential associated with S is /(/i, V, T) = -kBTlnS(/i, Vy T) (1.45) and shows that the grand canonical ensemble is a pi-V-T ensemble. The basic properties of the macromolecule can be derived from differentiation once S or / are known. The number of water molecules bound is )
(1.46)
Note that W is the same as N interpreted in the thermodynamic sense as an ensemble average. The pressure is (BJ P= (1.47)
dv
Finally, the derivative with respect to T yields J E -
= knr-
3 In 2
(1.48)
\
The derivative of the potential / with respect to T is again related to the entropy of the macromolecule in a direct way, as seen for the canonical ensemble. In fact, (dj\ JdlnE\ = J- E + = _k {n~_k = -S (1.49) Bn dT JHV \377/i,v " I dT c,v T Arguments similar to those used for the canonical ensemble show that the definition of S for a grand canonical ensemble is M
(1.50)
jV=O/=l
The Pfaffian form associated with / is dfi/v,T
(—)
dT= -Wd/i-
PdV-SdT (1.51)
and the integral form derived from eq (1.49) is J= E-TS-Wi*= F-Wn (1.52) The results of the foregoing analysis obtained for the microcanonical, canonical and grand canonical ensembles are summarized in Table 1.1.
1.1 Postulates and basic ensembles
17
Table 1.1. Summary of ensembles. Ensemble Microcanonical Canonical Grand canonical
Partition function Q(E, V, N) Z(N, V, T) SO*, V, T)
Potential 5 = kB In Q F= -kBTlnZ J= -kBT\nE
Pfaffian form dF dJ
SdT- PdV + [idN SdT - PdV -
Analysis of the properties of the macromolecule as a thermodynamic system entails the definition of a set of independent variables that can be controlled by external constraints. In the microcanonical ensemble the energy, volume and number of water molecules are fixed. In the canonical ensemble the energy fluctuates and can be defined only as an ensemble average over many distinct energy states or levels. In the grand canonical ensemble both the energy and the number of water molecules bound are subject to fluctuations. The fluctuating variable is defined in thermodynamic terms by computing the response of the partition function of the system, or equivalently the thermodynamic potential associated with it, with respect to the variables held fixed by external constraints. For instance, the change of F or / with T gives information on the entropy of the macromolecule. The change of / with [A provides the average number of water molecules bound. The properties of the macromolecule are thus derived from the response of the system to changes of temperature and the chemical potential of water. The properties of most interest in biochemical and biophysical studies are extensive quantities such as S, £ , V or JV. These quantities are derived as response functions of the system to conjugate intensive quantities such as T for S and £ , P for V and \i for N. Investigation of the properties of the macromolecule thus coincides with setting some external constraints that can be controlled experimentally, such as T9 P o r / i , and then measuring the response of the system. The basic ensembles dealt with in the foregoing analysis can be considered in this perspective. In the microcanonical ensemble all the properties of interest, i.e., the extensities E, V and N, are fixed. In this case it is not possible to measure any of these properties from fluctuations arising from changes in thermodynamic driving forces. The partition function is merely a constant that enumerates the configurations compatible with the preassigned values of E, V and N. In the canonical ensemble only V and N are fixed. In this case it is
18
1 Statistical thermodynamic foundations
possible to derive the energy of the macromolecule from the response of F to temperature changes. The canonical ensemble provides the simplest case of a system where a basic extensive property of the macromolecule is obtained as an ensemble average. In the grand canonical ensemble only V is fixed. Here both the energy, E, and the number of water molecules bound, N or W, are derived from ensemble averages. Both E and N are inherently fluctuating quantities to be derived as response functions to changes in T and /i. Clearly, the grand canonical ensemble is the system that most closely approximates the conditions usually encountered in practice. In general, it is very difficult to control extensive quantities pertaining to the macromolecule, while it is relatively easy to control temperature, pressure and the chemical potential of various components present in solution. For this reason, a thermodynamic description of the properties of the macromolecule must be based on suitable thermodynamic potentials, or partition functions, that are defined in terms of intensive quantities only.
1.2 The generalized ensemble
The basic ensembles dealt with so far provide a conceptual framework for the derivation of any potential of interest. Let us rewrite the integral definitions of the potentials F and / as follows F= E-TS (1.53) / = F - Afju = E - TS - Nfi (1.54) The potential F for the canonical ensemble is obtained from the energy E of the microcanonical ensemble by subtracting the term TS. Since E is a function of N, V and S only, the transformation (1.53) makes F a function of N9 V and T. If a similar transformation is applied to F by subtracting the term Np, then the potential / is obtained which is a function of //, V and T. This transformation is equivalent to subtracting TS and N{i from the energy E. Hence, any potential can be constructed from the energy E by replacing a given extensive quantity, S, V or N, with the conjugate intensity, T9 P or \i. The replacement is known as a Legendre transformation. The pair of quantities to be exchanged is subtracted from, or added to, the energy E to yield the new potential, depending on the sign of the pair in the Pfaffian form dE. Consider the differential of the energy dE = TdS - PdV + fidN
(1.55)
1.2 The generalized ensemble
19
which applies to the microcanonical ensemble. Assume that a new potential is to be derived from E where T replaces 5. Upon subtracting the differential d(TS) from both sides of eq (1.55) one has d£ - d(TS) = TdS - PdV + pdN - TdS - SdT = -SdT - PdV + pdN (1.56) The right-hand side of eq (1.56) is now a function of N9 V and T as desired, while the left-hand side is the differential of a new potential E — TS, which gives the Helmholtz free energy of a canonical ensemble. The transformation operated on E is, in fact, the same transformation that generates the canonical from the microcanonical ensemble. The potential for the grand canonical ensemble is derived from E by subtracting TS and NfA. In general, a total of 23 = 8 potentials can be generated with three independent variables. The first member of this set of potentials is the energy E, which is a function of extensive quantities only. As we have seen, some of the remaining potentials reflect properties of basic ensembles. Some have very little practical interest, considering the independent variables to be controlled experimentally. An example is the potential X = E — Np, which is a function of 5, V and /i. There are two potentials, however, of considerable practical interest since they refer to the properties of the macromolecule when pressure and temperature are set by external constraints. The first potential is the Gibbs free energy of the macromolecule obtained from E by interchanging P and V, and T and 5, i.e., G = E - TS + PV (1.57) The Gibbs free energy is a function of N, P and T and the N-P-T ensemble associated with it is called mechano-thermal. The Pfaffian form associated with G is evidently dG = -SdT + VdP + fidN (1.58) The partition function of the mechano-thermal ensemble can be derived in a way analogous to that considered for the canonical and grand canonical ensemble. The transformation which produces the desired potential from the energy E maps into a corresponding transformation involving the partition function of the system. The canonical partition function is derived from the microcanonical partition function by summing over all possible energy states. Likewise, summing over all possible energy states and values of N yields the grand canonical partition function. Hence, the change of a given extensive quantity with its conjugate intensity is
20
1 Statistical thermodynamic foundations
achieved by summing over all possible values allowable to the extensive quantity of interest. This turns out to be the result of finding the most probable distribution using undetermined multipliers, as shown in Section 1.1. To obtain the partition function for the mechano-thermal ensemble from the microcanonical ensemble, we have to sum over all energy states at constant T9 and over all possible volume states at constant P. Due to the continuous nature of the V variable, the sum over V must be replaced by an integral from the lowest (Vmin) to the highest (Vmax) value of V. Mathematically, the same result is obtained by summing from zero to infinity if we assume that macromolecular configurations with V < Vmin or V > VmSiX have vanishingly small probabilities of existence and make no contribution to the integral. The partition function Y(P, T, N) for the mechano-thermal ensemble is
= fz(V, T, A0exp(- ^-)dV Jo
\
kBTj
(1.59)
and is mathematically identical to the Laplace transform of the canonical partition function in V space. Again, the connection between Y and G is of the form G(N, P, T) = -kBTlnY(N, P, T) (1.60) The macromolecule as a mechano-thermal ensemble is indeed a mechanothermal device that can assume a number of different energy and volume states. This is exactly the scenario of interest in biology. The Gibbs free energy refers to the ensemble of such states when the number of water molecules bound is held fixed. The set of response functions derived from the potential G, or equivalently from the partition function Y, gives information on some important extensive properties of the macromolecule, such as V and E. The relevant expressions are \ 3P )N,T
E + PV = kBT4d-^]
\ 3T JN.P
(1.61)
\dP/ N,T
= -T:
(1.62) N,P N,P
1.2 The generalized ensemble
21
, JdlnY\ ( 3G\ n ,a. li=-kBT\ = (1.63) \ 3N )P,T \dN JP.T The important new feature in the mechano-thermal ensemble is that the volume of the macromolecule can be derived from the response of the system to pressure. The response to temperature is E + PV. Using the foregoing arguments this response can be considered as a new potential where V has been replaced by its conjugate P as an independent variable. The potential H = E + PV (1.64) is the enthalpy of the macromolecule and describes the properties of an N-P-S ensemble. The relationship in eq (1.62) is an important thermodynamic expression known as the Gibbs-Helmholtz equation, which relates the enthalpy to a change in the Gibbs free energy of the system. Combination of eqs (1.57) and (1.64) yields the key thermodynamic relation G = H - TS (1.65) from which any of the three thermodynamic quantities G, H and S can be derived from knowledge of the other two. We also have the property, analogous to eqs (1.17) and (1.49), that (dG\
, , , JdlnY\ G-E-PV /1 ^ = -kBlnYv - kBT[ = = -Sc (1.66) \dT/N,p \ dT )N,P T which complements the Gibbs-Helmholtz equation. The second potential of interest is obtained from E by interchanging all extensive variables with their conjugate intensities, i.e., n = E - TS + PV - Nil
(1.67)
We shall see that this potential, to be referred to as the generalized potential n , coincides with the chemical potential of the macromolecule /V- The potential II is a function of all intensive quantities \i, P and T and the [i-P-T ensemble associated with it is called generalized. The Pfaffian form associated with n is d n = -SdT 4- VdP - Nd/i
(1.68)
The partition function *P(P, T9 JA) of the generalized ensemble can be derived from that of the microcanonical ensemble by summing over all possible energy and volume states, as well as over all possible values of N. The solution is evidently
>, T. „) = f t 2 =xp I""-™-*"•*>] J
L
kT
J
iV
(1.69)
22
1 Statistical thermodynamic
foundations
with
no*, p, r) = -* B rinvoi, p, r)
(1.70)
Again, it is quite instructive to consider two alternative forms of W. One is obtained by rewriting W in terms of energy levels and their degeneracy as follows
T, ,) - f i s °WV. JOJ=IN=O
(1.71) This expression reveals the 'anatomy' of a generalized ensemble. The original microcanonical kernel Q is subject to a series of transformations leading to stepwise substitution of each extensive quantity with its conjugate intensity. Each substitution is brought out by filtering Q with an exponential term containing the thermodynamic pair of interest. Grouping terms in eq (1.71) according to a particular choice of independent variables generates various equivalent forms of W. The form of considerable practical interest is a polynomial expansion in the water activity w, as seen for the grand canonical ensemble, i.e., M
W(P, T,w)=^
Y(N, P, T)wN
(1.72)
The coefficients of this expansion are the mechano-thermal partition functions related to the Gibbs free energy of the macromolecule. In the expansion of the generalized ensemble, each intermediate state of the macromolecule defined by a given number of water molecules bound is a mechano-thermal device that can assume a number of different energy and volume states. Each intermediate enters the generalized partition function with a weight set by the Gibbs free energy of the corresponding mechanothermal device. The whole ensemble is generated by expanding the number of mechano-thermal devices along the N coordinate. The generalized ensemble provides the ideal system for studying the properties of the macromolecule, or any other component of interest. The importance of this ensemble was first pointed out by Stockmayer (1950). All the intensive quantities are controlled by external constraints and the extensive quantities reflecting the behavior of the macromolecule are determined from the response of II to thermodynamic driving forces such as P, T and \i. We have for these quantities the set of relationships
v = -t,i{***-) -M V SP ),.r
(1.73) WL
1.2 The generalized ensemble
23
=-T2
(1.74) \ 31nW (1.75) N-kBT -(-) \ dp ) I P,T \OUjP,l The change of n with T gives the entropy of the macromolecule, as expected PV -
dT =
_S
The entropy is related to the probability ipj(N, V) of observing a particular configuration of the macromolecule with preassigned values of N9 V and Ej. The expression for ipj(N, V) derived from application of the method of undetermined multipliers to the generalized ensemble is analogous to that already obtained for the canonical and grand canonical ensembles. The analytical form of ipj(N, V) is given by the ratio
Pi{N, V) =
I Y Y Q(N, (1.77) Of particular importance is to cast eq (1.77) in terms of the probability tyN of finding the Nth ligated intermediate with TV water molecules bound to it, i.e., YjN,P,T)w" M
N=0
T)N
_YjN,P,T)w" W(w, P,T)
The entropy of the macromolecule assumes a relatively simple form in terms of the xps M
S = -kB2VNlni>N
(1.79)
N=0
This relationship and eqs (1.73)-(1.75) provide the analytical tools to define the important extensive quantities of the macromolecule under consideration.
24
1 Statistical thermodynamic foundations
The importance of the generalized ensemble and the potential II in the description of the properties of the macromolecule becomes even more evident once the exact nature of II is clarified. This demands an explicit definition of the energy E. We have given the Pfaffian form of the energy in eq (1.29), but the definition of II, or G, H, F and any other potential, requires the definition of E in its integral form. Let us rewrite eq (1.29) as follows d£ = TdS - PdV + iidN + ^dN m
(1.80)
where the suffix m refers to the macromolecule. Since our system is the macromolecule itself, Nm is constant and its differential is zero. The term jUmdiVm added to eq (1.29) to obtain eq (1.80) makes no contribution to dE and the results of the foregoing sections remain perfectly valid, as long as Nm is kept constant throughout. Specifically iVm = 1, since our system contains only one macromolecule. Assume now that the value of Nm is multiplied by an arbitrary constant A> 1. This results in an expansion of our system. All other extensive quantities increase by the same amount A, since we have to take into account AN water molecules bound, a volume AV, an entropy XS and an energy XE. The expansion, however, is inconsequential on the properties of our system if we define a new set of variables scaled by a factor A with respect to the original extensities in eq (1.80). This leads to the important condition for the integral form of the energy E (Stanley, 1971; Hankey and Stanley, 1972; Weinhold, 1975a,b) E(XS9 XV, XN, XNm) = XE(S, V, N, Nm)
(1.81)
The energy of a system where all extensities have been scaled by a factor A is A times the energy of the original system with unsealed variables. Mathematically, eq (1.81) states that the energy E is a first-order homogeneous function. It then follows from Euler's theorem on homogeneous functions (Pearson, 1990) that £ = Vp^
Xi
(1.82)
where Xt denotes one of the t independent extensive quantities defining E. The partial derivatives in eq (1.82) can be calculated from the Pfaffian form (1.80) so that Nmiim
(1.83)
which gives the integral expression for the energy E. It is now easy to compute any other potential from eq (1.83) to obtain
1.2 The generalized ensemble
25
H = E + PV = TS + Nil + ^
(1.84)
F= E - TS = -PV + Nil + j v
(1.85)
J= E-TS-
Ntx= -PV + ^
(1.86)
G = £ - 73 + PV = Nil + ^
(1.87)
n = E - TS + PV - Nil = /v
(1-88)
where it should be remembered that Nm = 1 by construction. Hence, from the integral definition of E, we have proved that the potential of the generalized ensemble, n , is the chemical potential of the macromolecule itself. The Pfaffian form involving all intensive quantities of the system is SdT - VdP + Ndii + d/v = 0
(1.89)
Unlike the energy E, which is a function of all extensive quantities, the potential generated from E in eq (1.83) by replacing all extensities with their conjugate intensities, including the variables referring to the macromolecule, does not exist. Since P, T, /i and / v are all the variables to be taken into account in the system we have constructed, it follows that it is not possible to define a potential, and hence the state of a system, in terms of intensive quantities only. The relationship (1.89) that encapsulates this fundamental principle is the Gibbs-Duhem equation. Generalization of eq (1.89) to the case involving an arbitrary number of components is straightforward. The physical significance of the Gibbs-Duhem equation is as follows. Our two-component system cannot be uniquely defined in terms of P, T and the activity of water and the macromolecule. A solution of the macromolecule of a given activity (or concentration), at atmospheric pressure and 298.15 K does not define a thermodynamic system to which a potential can be associated. Knowledge of these conditions by no means allows us to derive how many water molecules are bound to the macromolecule (N), how many macromolecules are present (Nm), or the volume and energy of the system. To obtain this information we need to fix at least one of the extensive quantities involved, which is equivalent to specifying the size of the system. All the potentials defined in the foregoing analysis, except n , contain the term dNm in their Pfaffian form. Since Nm is fixed, dNm makes no contribution and all potentials are functions of three independent variables. However, the importance of fixing Nm is revealed by the fact that its conjugate variable, jMm, itself becomes a member of the set of potentials, and is itself a function of three independent variables. Thus, fixing one extensity results in the generation of a set of potentials that is iso-homogeneous. If one of the extensities is
26
1 Statistical thermodynamic
foundations
not fixed, then there are potentials such as E containing four independent variables and a null potential corresponding to the Gibbs-Duhem equation. In a system containing N extensive variables and a corresponding number of conjugate intensities, there can only be TV — 1 independent intensities, as dictated by the Gibbs-Duhem equation. Hence, fixing one of the extensities is a necessary condition to define the state of the system. This operation yields a total of N - 1 independent variables, extensive or intensive.
1.3 The laws of thermodynamics
The generalized ensemble depicts the macromolecule as an open, isothermal system at constant pressure. These are the conditions in which macromolecules function in vivo and for this reason they are of most interest in practical applications. As a result of the properties of the generalized ensemble, thermodynamic driving forces act upon the system and modify its state in a way that can be predicted from the analytical behavior of a set of response functions. These driving forces may be expected to interfere with each other if the response of the macromolecule, say to P, modifies its response to T or /i. Thermodynamics provides a number of rules for the interference to take place. These rules are at the basis of many important regulatory phenomena in biology and prove useful for understanding and developing basic mechanisms of action of macromolecules under equilibrium conditions. The Pfaffian form of the energy for the system of interest composed of the macromolecule and the solvent embodies the results of the first two laws of thermodynamics. The first law establishes the conservation of energy and the fact that for any reversible transformation the change in the energy E depends solely on the initial and final states, as implied for a state function. The second law establishes that, when all other extensive variables are fixed, the energy of a system at equilibrium is at a minimum and that any perturbation of the equilibrium state triggers a response from the system to restore the initial conditions. We shall now examine the important consequences of these two laws. Mathematically, the first law states that dE is an exact differential, since E is a state function. Let us write the familiar Pfaffian form dE = TdS - PdV + ^dN (1.90) in terms of the three independent variables S, V and N, as done in Section 1.1. We maintain that Nm is constant and that the macromolecule
1.3 The laws of thermodynamics
27
is our system by construction. Euler's theorem on exact differentials (Wilson, 1911; Pearson, 1990) gives rise to a number of relationships between different independent variables. In the case of dE the reciprocity relationships are N,V
) ,s 3N fv,s
id) 3VJN,S
\dS/N,
.J*L) \3N
(1.93) )S,V
These relationships are also known as Maxwell's relationships. A similar set of relationships exists for any of the potentials derived in the foregoing sections, since all of them are state functions like E and have exact differentials. Here we choose n due to the importance of the generalized ensemble in the description of the properties of the macromolecule. The Pfaffian form can be rewritten in a particularly convenient form as follows - d n = SdT - VdP + Ndfi (1.94) The exact differential — d n yields a form similar to dE, where all extensive independent quantities are replaced by their conjugate intensities. The Maxwell's relationships associated with —II are
(1.96)
We will discuss these relationships in order. The first relationship connects the entropy and the volume of the macromolecule. The change in entropy due to a change in pressure, while keeping /i and T constant, is the same as the change in volume (with opposite sign) due to a change in temperature, while \i and P are constant. Hence, if pressure affects the entropy of the macromolecule at constant T and \i, then temperature must affect the volume of the macromolecule at constant \i and P. In this case entropy and volume are linked. On the other hand, if pressure has no effect on the entropy, then temperature cannot affect the volume of the macromolecule. The second relationship connects entropy and the number of
28
1 Statistical thermodynamic
foundations
water molecules bound. If a change of the entropy of the macromolecule is observed upon a change of the chemical potential of water, then temperature must affect the amount of water bound to the macromolecule. Entropy and TV are linked. The last relationship applies to linkage effects involving N and V. If pressure affects the number of water molecules bound to the macromolecule, then a change in the chemical potential of water must affect the volume of the macromolecule. We begin to perceive a rather basic and important aspect of the thermodynamics of macromolecular systems from the Maxwell relationships derived for the potential II. Different driving forces acting on the macromolecule, such as P, T or //, may bring about changes in the properties as revealed by specific response functions. These changes are not arbitrary, but obey some important reciprocity rules. Consider an extensive property such as N9 the number of water molecules bound to the macromolecule. The response of N to driving forces such as T or P, that are not conjugate to N, provides information on how \i, the driving force conjugate to N, will affect S or V, the extensive quantities conjugate to T and P. Therefore, one can use a particular response function to predict others that are difficult to access experimentally. For example, if N is measured as a function of P and a change is observed, then it can be concluded that the binding of water will affect the volume of the macromolecule. This is a very useful result, considering that measurements of V are not as direct and feasible as measurements of N. Likewise, assume that we are interested in the effect of \i on the entropy of the macromolecule, but there is no direct way of measuring the effect. Measurements of N as a function of T will reveal whether or not a change of S due to a change of \x is to be expected. The second law establishes a sort of variational principle of substantial importance for thermodynamic response functions. The first law establishes a connection between transformations involving pairs of conjugate variables through the Maxwell relationships, but it says nothing about the direction of the change. According to the first law, any transformation is allowed to take place, as long as conservation of energy is not violated. The second law imposes further restrictions and adds predictive power to the thermodynamic approach. This law states that, for a given set of extensive variables held fixed, only those transformations that minimize energy can take place in a thermodynamic system at equilibrium. Mathematically, the first two laws can be expressed in a simple and most elegant way using matrix algebra (Gibbs, 1928; Weinhold, 1975a,b). Consider the definition of the energy function E in terms of a set of independent
1.3 The laws of thermodynamics
29
extensities {Xi} = Xi, X2, . . . Xr and let fi be the intensive variable conjugate to Xt. Consider the r x r matrix G with elements (/,/ = !, 2, . . . r)
(1.98)
where the suffix {Xk±j} denotes that all Xs but Xj are kept fixed. The first law states that the matrix G is symmetric. In other words, the equality gij = gjt
(L99)
gives the Maxwell relationships in terms of the elements of G. The second law provides additional information on the properties of G. The energy E is a minimum if, and only if, the matrix G is positive-definite (Gantmacher, 1959). This implies that all principal minors of G are positive. A principal minor is a determinant obtained from G by deleting rows and columns crossing over a diagonal element. In the specific case of our system, we have r = 3 and we can identify 5 with Xu V with X2 and N with X3. Hence, f1=Tyf2 = -P and f3 = [i. The matrix G is
(
gll gll
gl2 guX g22 g23
g31
g32 #33/
(1.100)
The positive-definite nature of G implies that all of the following conditions are satisfied gn>0 (1.101a) g22>0 (1.101b) g33>0 (1.101c) (1.102a) gngn ~ gngn ->0 (1.102b) gllg33 ~ gl3g31 '>0 (1.102c) g22#33 - g23g32 '>0 gllg22g33 + gl2g23g31 + gl3g21#32 - gl3g22#31 ~ gl2#2lg33 ~ gllg23g32 > 0
(1.103) The inequalities (1.101a)-(1.101c) establish that the diagonal elements of G are always positive. A diagonal element is by definition the response of a given intensive variable to a change in its conjugate extensity. Hence, an increase in entropy implies that temperature has increased, since gii = dT/dS > 0. Likewise, an increase in the number of water molecules bound implies an increase in the chemical potential of water, since £33 = 3ju/3iV > 0. Finally a decrease in pressure is linked to an increase in the volume of the macromolecule, since g22 = —dP/dV > 0.
30
1 Statistical thermodynamic
foundations
While the diagonal elements of G involve pairs of conjugate variables, the off-diagonal terms reflect the mutual interference, or linkage, of different independent variables. In addition to the criterion involving diagonal elements, the second law defines constraints also for the offdiagonal elements, or linkage coefficients. The inequalities (1.102a)(1.102c) demand that for any two pairs of conjugate variables the product of the diagonal elements be larger than the product of the linkage coefficients. Similar restrictions apply to all other principal minors, including the determinant of G in eq (1.103), which is itself a principal minor. Other conditions embodied by the second law are derived from consideration of the potential IT as a function of the independent intensities {/i} = fu fii • • • fr» The r x r matrix G' associated with —II has elements . ^
v _,
f
j
«,_,....
y
(1.104)
For our system of interest G' is given by G' = \g'21 \
g22
g^\
(1.105)
£33
Again, the first law states that the matrix G' is symmetric, so that g,y = g'j h while the second law imposes on G' a set of conditions analogous to eqs gii > 0
(1.106a)
g22>0
(1.106b)
g33 > 0
(1.106c)
g!lg22 " gl2g21 > 0
(1.107a)
gllg33 - g!3g31 > 0
(1.107b)
g22g33 " g23g32 > 0
(1.107c)
gllg22g33 + gl2g23g31 + gl3g2lg32 ~ gl3g22g31 ~ gl2g2lg33 ~ gllg23g32 > 0
(1.108) As in the case of G, we have for G' conditions involving diagonal elements and off-diagonal linkage coefficients. The diagonal elements of G' play a fundamental role in the thermodynamic description of the system. The response of 5 to T, at constant P and j« -) =- ^ (1.109) KdTJp* T gives the heat capacity of the macromolecule, divided by the absolute
1.3 The laws of thermodynamics
31
temperature. The heat capacity is a measure of the heat absorbed reversibly by the macromolecule due to an increase in temperature. The second law demands that the heat capacity is always positive. Hence, the macromolecule absorbs heat when T increases and releases heat when T decreases. A response function analogous to the heat capacity is the binding capacity (Di Cera, Gill and Wyman, 1988a) 3 ^ )P,T
which represents the change in the number of water molecules bound due to a change in the chemical potential of water. The second law states that the binding capacity is always positive and that an increase/decrease of the chemical potential of water always increases/decreases the number of water molecules bound. The third diagonal element gives the change of V with respect to P, at constant T and \i
and represents the compressibility of the macromolecule times the volume. The compressibility is always positive, since an increase/decrease in pressure always decreases/increases the volume of the macromolecule. The second law guarantees the stability of the equilibrium state against small, adiabatic perturbations. Assume, for instance, that the equilibrium of the system is perturbed by a slight increase in pressure. This change causes the volume of the macromolecule to decrease. The response of the macromolecule relieves the external 'stress' induced by the perturbation and restores the original conditions. The pressure is, in fact, inversely proportional to the volume of the system not occupied by the macromolecule and the solvent (outer volume), so that a decrease in the volume of the macromolecule opposes the increase in pressure by increasing the outer volume. If the second law were violated and the macromolecule were to react by increasing its volume, the outer volume would shrink and produce a further increase in pressure. The sequence of reactions triggered by the anomalous response of the macromolecule would then make the pressure grow without bounds in a spontaneous fashion, thereby leading to explosion of the system. Similar arguments apply in the case of the response of the macromolecule to T and \i. An increase in temperature causes the macromolecule to absorb heat, thereby minimizing the rise of temperature. An increase in the chemical potential of water causes water molecules to bind to the macromolecule. If the number of water molecules bound were to decrease with increasing JM, thereby violating the
32
1 Statistical thermodynamic
foundations
second law, a small increase of pi arising from a spontaneous fluctuation would eventually cause all bound water molecules to dissociate from the macromolecule. A small decrease of pi would cause all water molecules in the bulk solvent to condense on the macromolecule. Thermodynamic stability, on the other hand, makes the macromolecule act as a buffer that absorbs water when pi increases and releases water when pi decreases, in either case opposing the effect of the perturbation acting on the system. The response to T is analogous, so that heat is absorbed/released when T increases/decreases. The natural tendency of the system to react to the perturbation and restore the original conditions of equilibrium is the mark of thermodynamic stability and forms the basis of Le Chatelier's principle on the restorative tendency of the equilibrium state (Prigogine and Defay, 1954). In addition to the diagonal elements of G and G', other principal minors are subject to the constraints imposed by the second law. It is quite instructive to show that these minors are analogous to the response functions defined above. Consider the set of exact differentials involving the extensive quantities of the macromolecule under consideration
M
( ] \ ST )P,V
dT + (
\ 3P JT,P
dP + l ) \ dpi JP,T
(1.113) d,
(1.114)
This set is equivalent to the transformation dXx = g'n d/i + g[2 d/2 + g[3 d/3
(1.115)
dX2 = g'2l d / i + g'22 d/ 2 + g'2, d/ 3
(1.116)
dX3 = gh d/x + g',2 d/2 + g^3 d/3
(1.117)
that yields the set of extensities from the set of intensities through the matrix G'. The transformation exists and is unique since the second law guarantees that the determinant of the matrix G' is positive (Gibbs, 1928; Shaw, 1935; Crawford, 1950, 1955; Weinhold, 1975a, b). Assume now that the variable X3 is held fixed, so that dX3 = 0. This allows d/3 to be substituted in eqs (1.115) and (1.116) using eq (1.117) to yield a new transformation
13 The laws of thermodynamics
^
33
(1.118) (1.119)
The coefficients of this new transformation define a 'contracted' form 3 G' of the original matrix G', and denote response functions subject to the condition X3 = constant. For example, 3 gn denotes the change of S with T when P and N are kept constant, as opposed to g'n that represents the analogous change when P and \i are constant. Likewise, 3g22 is the change of V with — P at constant T and N, as opposed to g22 that defines the same change at constant T and /a. The contraction of the matrix G' along the variable X3 = N has operated the substitution f3 —> X3 in the definition of the response functions involving the thermodynamic pairs fl9 Xx and / 2 , X2- The connection between the elements of 3 G' and those of G is g33
Since the matrix 3 G' has been derived from the matrix G' by fixing the state of one thermodynamic variable in the system, it is itself subject to the first two laws and is itself symmetric and positive-definite. It follows from the first law that 3g{2 = 3 g 2 i, or that
1?)
3PJN,T
\C
/P,N
The only difference between the Maxwell relationship above and eq (1.95) is in the independent variable N which replaces iu. The following set of inequalities is a consequence of the second law
3
g22 = -(—) 3
gn3g22 -
=KTfNV>0 3
g'n3g2i > 0
(1.123) (1.124)
The heat capacity and compressibility of the macromolecule retain their fundamental property of being positive. This result is a straightforward consequence of the properties of G'. In fact, eq (1.120) implies that 3g'n is the ratio between two principal minors of G \ i.e., 3 , _ gllg33 ~ gl3g31 gn g33
/ 1 10 ^x (L.LZD)
that are themselves always positive. The same argument applies to 3g22. It also follows from eq (1.125) that g n ^ 3 g i i , since the product g(3g3i is
34
1 Statistical thermodynamic foundations
always positive due to the Maxwell relationships. Hence, CP,^CPfN (1.126) The heat capacity of the macromolecule computed at a constant chemical potential of water is never smaller than the heat capacity computed at a constant number of water molecules bound. If the temperature is increased/decreased at constant \x the heat absorbed/released by the macromolecule is never smaller than that absorbed/released while keeping TV constant. Likewise, *T> ^ KT,N (1.127) The compressibility of the macromolecule computed at constant \i is never smaller than that computed at constant N. Had we chosen the condition X2 = constant in (1.116) we would have obtained Cp,p & CVtll (1.128) BP,T^BVT (1.129) Likewise, letting Xx = constant in eq (1.115) leads to KT* ^ *s# (1-130) BptT^BPtS (1.131) The inequalities (1.128) and (1.130) provide more information on the heat and volume changes of the macromolecule. The binding capacity is subject to similar constraints and at constant T or P is never smaller than the binding capacity computed at constant S or 7 . In general, for any response function expressing the change in a given extensity due to a change in the conjugate intensity, the change computed at constant / , where / is an arbitrary intensive quantity, is never smaller than that computed at constant X, where X is the extensity conjugate to / , or g'u^tgu (L132) where j' + i. It is easy to verify by induction that the same inequality holds for the case where an arbitrary number of extensities is kept constant.
1.4 Fluctuations
It is quite instructive to note that the elements of the matrix G', as well as those of G, are related to the fluctuations of important thermodynamic quantities. Consider the polynomial expression of the partition function W for the generalized ensemble (1.72). The first derivative of ln1^ with respect to In w, at constant P and 7, is equivalent to differentiation with
1.4 Fluctuations
35
respect to \i. The result is the average number of water molecules bound, W, as follows M
W
M
\ 3 In w )P,T
;=0
Here / replaces the index TV in eq (1.72), while Aj = Y(y, P, T) is the mechano-thermal partition function for the configuration of the macromolecule with j water molecules bound. The derivative of W with respect to lnw, dW/dlnw, at constant P and T, gives the binding capacity B. This is a simpler definition than that given in eq (1.110), since the kBT term in the denominator has been omitted. This term is, of course, inconsequential when T is constant. The relevant expression written in terms of the partition function of the generalized ensemble is obtained by differentiation of eq (1.133) as follows M
2
31nw /^
i M
j=o
;=o
M
M
;=0
i=o M
I M
={J2)-{i)2 ;=0
\;=0
(1-134)
/
The binding capacity is therefore the variance of the distribution of ligated intermediates around the mean value W defined by eq (1.133), or a measure of the fluctuations associated with the average value W. f Similar arguments show that the compressibility is related to the fluctuations of the volume. Upon expressing W as the Laplace transform of the grand canonical partition function of the system = fexp(-PV/kBT)Z(V, Jo
T, ii)dV
(1.135)
The connection between the derivative of W and the variance of the fluctuations of the molecules bound was presented by Linderstr0m-Lang at a lecture at Harvard in 1939 (Cohn and Edsall, 1943; Wyman and Gill, 1990) and believed by many to be derived by him originally. Tolman (1938) points out that eq (1.134) was first derived by Pauli in 1927 (Pauli, 1927). It turns out, however, that eq (1.134) is as old as statistical mechanics and was first published by Gibbs in 1902 (Gibbs, 1902). The derivation of eq (1.134) is also implicit in Gibbs' monumental paper on the equilibrium of heterogeneous substances (Gibbs, 1875).
36
1 Statistical thermodynamic
foundations
the volume is derived as
fvcxp(-PV/kBT)E(V,
Jo v
/vj3 J- I
fexp(-PV/kBT)E(V,T,fi)dV Jo
(1.136)
onstant i ana filS 1 VkBT*
V
pvn
PV \^ kBTj
1
PV \^ I Jo
k BT )
(1.137)
fexpl-fZ- S . Jo \ /c B r/ and depends on the variance of the distribution of volume values. Likewise, the heat capacity is the variance of the fluctuations of the energy. The foregoing relationships identify important response functions of the macromolecule as fluctuations of thermodynamic extensive quantities and are equilibrium analogues of 'fluctuation-dissipation relationships' (Kubo, 1965; de Groot and Mazur, 1984). The conditions imposed by thermodynamic stability on the sign of these response functions are readily understood in terms of statistical considerations, since the variance of a distribution is necessarily a positive quantity. The linkage coefficients, on the other hand, are related to the covariance of the fluctuations of extensive quantities. For example, the change of W with respect to P is derived from eq (1.133) and W given in eq (1.72) as
Jsw
(1.138)
\ dP The change of V with respect to \i yields k BT
The Maxwell relationship (1.97) has a straightforward interpretation in terms of the covariance of the fluctuations in the number of water
1.5 Simple models for energy transition
37
molecules bound and the volume of the macromolecule. The covariance of two variables a and b is, in fact, the same as the covariance of b and a (Feller, 1950). The matrix G', as well as its inverse G, is the variance covariance matrix, or Hessian, of the system defined in terms of the independent thermodynamic variables (Feller, 1950). Thermodynamic stability imposes on G' the quite obvious restriction of being positive-definite as is necessarily the case for a Hessian matrix. Alternatively, the matrix G', or its inverse, can be seen as the Gram matrix (Gantmacher, 1959) of a Euclidean hyperspace spanned by t independent vectors, each of which is associated with the differential of an intensive quantity. In this metric picture, elaborated by Weinhold in a series of most elegant papers (Weinhold, 1975a, b, c), each response function becomes the projection of a given vector onto itself (diagonal elements of the Gram matrix), or another vector of the basis set (off-diagonal elements of the Gram matrix). Thermodynamic laws translate into rules of Euclidean geometry. The projection of any vector onto itself is necessarily a positive quantity, hence the positive nature of the binding capacity, compressibility and heat capacity. On the other hand, the projection of a vector a onto a vector b equals the projection of b on a, hence the Maxwell reciprocity relationships involving linkage coefficients. The positive-definite nature of G' is a direct consequence of the positive-definite metric of the Euclidean space associated with it. The determinant of G' is related to the volume of the hyperspace and must be positive. Any principal minor of G' represents the volume of a subspace orthogonal to a given vector of the basis set and must be positive for the same reason.
1.5 Simple models for energy transition
We shall now examine a number of specific models that are relevant to macromolecular interactions. These models will be useful for our subsequent analysis of cooperative transitions. We start with the simplest model of interest by considering the macromolecule as a canonical ensemble, with the volume and number of water molecules bound fixed to arbitrary values. The canonical partition function of the system is given by eq (1.30)
Z{T) = 2 O ( £ y ) e x p ( - ^ y y=i
\
(1.140)
kBTj
where TV and V have been omitted since they are assumed to be constant. We assume that the macromolecule can exist in two energy levels, — e and
38
1 Statistical thermodynamic foundations
+ £, each of which has a degeneracy of one. The partition function is in this case
The Helmholtz free energy, the energy E and the entropy 5 of the macromolecule are given by F= -kBTln2=
A:BTlncosh(—— |
(1.142)
U77
_£tanh(-±-)
(1.143)
U CBJ 7 7 B
S = A:Bln2 + A;Blncosh
—
(1.144)
\kBT) T The energy E is derived from F using eq (1.14). The important response function of the system is the heat capacity at constant V obtained from E by differentiation with respect to T, i.e., £
Cv =
=- ^ (1.145) kB T2 cosh2 (e/kB T) cosh2 0 where © = e/kBT is a dimensionless variable. The heat capacity is illustrated in Figure 1.6 as a function of 0 . The function is positive everywhere and tends to zero for © —»0 and 0 —» oo ? corresponding to T ^ o o and T—>0 respectively. Since C v is a measure of the fluctuations of £ , the value of Cv goes to zero whenever E approaches asymptotically a constant value. As T—»0, the macromolecule freezes into the lowest energy level — e. As T—>°°, both energy levels become equally populated and the system freezes into another regime that is the average of the two possible energy levels. In either case, no fluctuations of the energy occur. It is intuitively obvious that at some intermediate values of T the macromolecule makes a transition between these two limiting regimes. The maximum value of Cv is observed for a value of 0 = ©max such that 0 max tanh0 max = l (1.146) thereby indicating that the macromolecule is making a transition between two energy levels. The transition occurs at a value of the energy E = -knT (I 147) ^max
^ B L max
\±'±^'/
A peak in the heat capacity profile of the system thus indicates the existence of at least two energy levels accessible to the macromolecule. The levels are differentially populated as the temperature increases. At low temperature only the level with minimum energy is populated, while, as temperature increases, levels with higher energy also become pop-
1.5 Simple models for energy transition
39
IT)
d
CO
d
CN
d
q d 0.0
1.0
2.0
3.0
4.0
5.0
0 Figure 1.6 Plot of the heat capacity, in kB units, versus the dimensionless variable 0 = e/kBT for the two-level model for energy transition. The peak in the function Cv is known as a Schottky transition and reveals the presence of a transition between two distinct energy levels.
ulated. A fundamental property of the system, i.e., the existence of multiple energy levels, is revealed by the analysis of the fluctuations of the energy using thermodynamic response functions. The heat capacity of the form (1.145) is also called the Schottky specific heat and is characteristic of any substance having an 'excitation' energy 2e (Kubo, 1965). The next model of interest is the case of a quantized energy spectrum. In this case the macromolecule modeled as a canonical ensemble can exist in an infinite number of equally degenerate energy levels Ej = Eo + je
(1.148)
with j = 0, 1, . . . oo. The canonical partition function is given in this case by
>(-E0/kBT) y=0
(1.149)
1 Statistical thermo dynamic foundations
40 Hence,
(1.150)
F = Eo + kB T In 1 - exp I -
E= S= Cv =
(1.151)
cxp(e/kBT) -
(1.152)
T[exp(e/kBT)-l]
£2exp(—e/k BT)
_
2
kBT [l - exp(-e/k B T)f
A:B02exp(—0) [1 - e x p ( - 0 ) ]
A: 2
B0 2
4sinh (0/2) (1.153)
The heat capacity profile is shown in Figure 1.7. A comparison with the
in
d
q d 0.0
1.0
2.0
3.0
4.0
5.0
0 Figure 1.7 Plot of the heat capacity, in kB units, versus the dimensionless variable 0 = e/kBT for the quantized energy model for energy transition. Unlike the two-level model, there is no peak in the function Cv which changes monotonically from zero to k&.
1.6 Simple models for binding
41
heat capacity profile in Figure 1.6 obtained for the two-level model shows the absence of a peak. The heat capacity for the quantized energy levels changes from a value of zero, obtained for 7"—>0, to a value of k B for T -> oo. This indicates that for T —> 0 the macromolecule freezes into a constant energy level, as in the case of the two-level model, but at higher temperature the energy increases in a linear fashion with T. In fact, from eq (1.151) it follows that
E~ Eo + kBT
(1.154)
for T » e/k B. The presence of fluctuations of the energy at high temperature values for the quantized energy model is to be understood in terms of the availability of arbitrarily high energy levels for the system. These levels, which are not populated at low temperature, become progressively stabilized as T increases, thereby causing the energy of the system to fluctuate constantly. 1.6 Simple models for binding We now consider the grand canonical analogue of the models treated in the previous section. These simplified models are conceptually useful in the discussion of binding phenomena. The macromolecule is modeled as a generalized ensemble for which P and T are fixed, while the number of water molecules bound is subject to fluctuations under the effect of the driving force \i. We start from eq (1.72) written as a polynomial in the water activity w as follows M
%)=XV
(1-155) N=O
which incorporates the mechano-thermal partition functions for given values of N. We now assume that the macromolecule has a single binding site for water, which can exist in two states, free (N = 0) or bound (N = M = 1). This situation parallels the two-level model for energy transitions. Likewise, we assume that the ligation state of the macromolecule uniquely specifies its Gibbs free energy level, Go for the site free and Gx for the site bound. The degeneracy of these levels is obtained from the dispositions allowed for the water molecule in all possible ligated configurations. Evidently, there is only one way to obtain the site, free or bound, and the microcanonical components of the partition function (1.71) are equal to one. Hence,
W(w) = expf——
)(1 + Kw)
(1.156)
42
1 Statistical thermodynamic
foundations
where
ft?) •
is the equilibrium constant for the reaction free —> bound of the site and is a function of T and P. This is a consequence of the law of mass action (Prigogine and Defay, 1954) through which water binds to the site on the macromolecule in a specific and 'saturable' way that can be predicted from knowledge of the water activity. The standard free energy of ligation for the site is (1.158) = -kBTlnK The partition function (1.156) is a binomial expression in the variable w, times a factor that depends only on temperature. This factor is inconsequential for the properties of the macromolecule if our discussion remains confined to an isothermal system. The quantities of interest in this system are the average number of water molecules bound, W, and its fluctuations or binding capacity B. Both quantities are obtained from differentiations of lnW as already shown in the foregoing sections. The explicit expressions for W and B are _ 31nW _ rr
Kw I 1.1J7
—
3 In w
I
1 + Kw
31nw {l + Kwf The functions W and B are plotted in Figure 1.8 versus the logarithm of the water activity. The value of W goes from 0 to 1, two asymptotic values obtained for w = 0 and w = <*>. The binding capacity is positive everywhere and shows a peak for wmax = K~l. This is the value of w for which W = \ and the macromolecule is equally partitioned between free and bound sites. The two-state model for the ligation of the site echoes the two-level model for energy transitions examined in the previous section. The driving force for binding is lnn>, while, in the case of energy transitions, it is the temperature T. The function W parallels E, while the binding capacity parallels the heat capacity. The physical significance of the binding capacity can be understood as a Schottky-like effect arising as a consequence of the transition between two ligation states. A peak in the binding capacity profile indicates the existence of such a transition reflecting the ligation of a site. The quantized analogue for binding is provided by a binding site that can accommodate an arbitrary number of water molecules. This situation,
1.6 Simple models for binding
43
00
c>
CO
d
£ *
c>
q 6 -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log w Figure 1.8 Plot of W (continuous line) and B (dashed line) versus the logarithm of water activity for the case of water binding to a single site of the macromolecule according to the law of mass action. Curves were drawn according to eqs (1.159) and (1.160) with K = 1 NT1.
which is of interest when dealing with protein hydration phenomena, entails binding of a water molecule to the site much like what is observed in the two-state model, followed by subsequent binding of additional water molecules in a piggy-back fashion, as illustrated in Figure 1.9. This case, to be referred to as condensation (Mayer and Mayer, 1940), is quite different from mass law binding since it does not show saturation. We assume that the Gibbs free energy of the configuration of the site with TV water molecules bound is given by GN = G o + NG
(1.161)
where N = 0, 1, . . . <*>. The analogy with eq (1.148) is obvious. The degeneracy of each free energy level is equal to 1, since there is only one way of ligating the first water molecule, or any water molecule thereafter
44
1 Statistical thermodynamic foundations
Figure 1.9 Schematic representation of mass law binding (top) and condensation (bottom), for the paradigmatic example of water binding to a macromolecule. In the case of saturable binding, each site is occupied by at most one water molecule. In the case of condensation, multiple layers are formed as vertical piles of molecules for each site. Saturation is the landmark property of mass law binding.
in a piggy-back fashion. The partition function can be written as (L162) 1 v 1 - Kw where K = exp(—e/k BT) and again the constant term exp(— G0/kBT) is inconsequential. The relevant response functions are Kw W = (1.163) 3 In w - Kw dW Kw B= (1.164) (1 - Kw)2 These expressions are the same as those derived for the case of mass law binding, except for the minus sign in the denominator. The replacement has the nontrivial consequence that both W and B are defined only in a finite range of w such that K-1 = wc> w^O (1.165)
The critical value wc defines the water activity at which all water molecules present in solution condense on the macromolecule, with a consequent divergence of W and B. The behavior of W and B is illustrated in Figure 1.10. The value of W grows without bound as H>—> wC9 and so does B. The saturation seen in the function W for the two-level model in Figure 1.8, which is the mark of mass law binding, is never reached in the model for condensation. Accordingly, the binding capacity does not peak for any value of w, since the system never settles down to a definite configuration upon binding and the fluctuations grow without bound as w increases.
1.6 Simple models for binding
45
o
CO
q CN
U
O
o d -3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
log w Figure 1.10 Plot of W (continuous line) and B (dashed line) versus the logarithm of water activity, for the case of water binding in a nonsaturable way (condensation) to a single site of the macromolecule. Curves were drawn according to eqs (1.163) and (1.164) with K = 1 M"1. The critical value of the water activity is wc = 1 M.
The two models analyzed above provide important examples of interactions. In the case of the two-state model, the statistics hinge on the transition between the free and bound states of the macromolecule. This situation is completely analogous to that arising in the Fermi-Dirac statistics of quantum mechanics, where any given quantum state can be occupied by at most one particle (Tolman, 1938; Hill, 1960; Kubo, 1965). In our case, the binding site of the macromolecule plays the role of the quantum state that can be either free or occupied by a particle to be identified with a water molecule. The macromolecule binding according to the law of mass action behaves like a 'fermion'. In the condensation model, there is no upper bound for the variable W and one or more water molecules can occupy the binding site. This case is completely analogous
46
1 Statistical thermodynamic foundations
to that arising in the Bose-Einstein statistics of quantum mechanics, where any given quantum state can be occupied by an arbitrary number of particles (Tolman, 1938; Hill, 1960; Kubo, 1965). The macromolecule binding according to the condensation model behaves like a 'boson'. The two models can be merged to describe high-affinity mass law binding to a given site, followed by low-affinity condensation. This hybrid model is known as the Brunauer-Emmett-Teller theory of adsorption and yields the simple Langmuir theory as a special case in the absence of condensation (Hill, 1960). Interesting extensions of this theory to the case of fractal layers of adsorption have been derived (Avnir, Farin and Pfeifer, 1984; Fleischmann, Tildesley and Ball, 1990). In the rest of this monograph, we will deal exclusively with mass law binding.
2 Global binding processes
In this chapter we start our discussion of binding processes. Here we deal explicitly with processes that involve the system as a whole. The properties of the macromolecule in this global description can be derived in a straightforward manner from the principles outlined in the previous sections.
2.1 The reference system
The simplest system of interest in the discussion of binding processes is that composed of a macromolecule M containing a single site for a ligand X. The term 'ligand' indicates a molecule that interacts specifically with M according to the law of mass action as discussed in Section 1.6. In the previous chapter we have considered water as ligand. Here we are interested in the interaction of M with a 'specific' ligand and regard the solvent as an 'inert' component. A treatment of linked hydration effects is given in Section 2.6. For the sake of simplicity in the following we consider temperature and pressure constant. We also assume that the macromolecule does not change its aggregation state. A rather extensive and elegant treatment of aggregation effects linked to binding processes has been developed by Hermans and Premilat (1975). The macromolecule is modeled as a generalized ensemble for which P and T are fixed, while the number of ligands bound is subject to fluctuations under the effect of the driving force \i = A;Brinjt, where x is the ligand activity. The relevant partition function of this system has the same form as eq (1.155), i.e., W(x) = Yo + YlX (2.1) The mechano-thermal partition functions, Yo and Yl9 incorporate the properties of the two possible ligated intermediates and specify the values 47
48
2 Global binding processes
of the related Gibbs free energies Go and Gi. The standard free energy difference between the bound and free forms of the site is AG = Gx - Go = -kBTlnK (2.2) where K is the equilibrium constant for the reaction M + X<=>MX (2.3) or, according to the law of mass action
K = IM^I
(2.4)
[M]* We implicitly assume that the solution is dilute enough to allow for the concentrations of the macromolecular forms M and MX to be used in place of their activities. The equilibrium constant K is a function of T and P, as expected for a mechano-thermal ensemble, and the change of K with T and P provides information on enthalpy and volume changes associated with the binding reaction (2.3), as implied by eqs (1.61) and (1.62). Detailed treatments of these effects are given elsewhere (Wyman and Gill, 1990; Weber, 1992). The degeneracy of Go and Gx is set by the dispositions allowed for the ligand in all possible ligated configurations. Evidently, there is only one way to obtain the site free or bound, which sets the microcanonical components of the partition function (2.1) equal to one. We now rewrite the partition function as W(x) = exp - — ^ (1 + Kx) \ kkBTjT)
(2.5)
B
At constant T and P the prefactor makes no contribution to the binding properties of the system and can be set equal to one without loss of generality. This is equivalent to setting Go = 0, or taking the unligated form of the macromolecule as reference. Under this assumption, the partition function assumes the rather simple form W(x) = 1 + Kx (2.6) The average number of ligated sites per macromolecule, X, and the fluctuations of X reflecting the binding capacity B, are obtained from differentiation of lnW as shown in Section 1.6. The result is dlnx
1 + Kx
B = -**- = ^ (2.8) 2 dime (1 + Kx) The partial derivative has been replaced with the total derivative for the
2.1 The reference system
49
sake of simplicity, since T and P are constant. The functions X and B plotted versus the logarithm of the ligand activity give the results already shown in Figure 1.8. Binding to the macromolecule shows saturation at X = 1 as x —» oo. The binding capacity shows a peak for x = K~l, which is the value of the ligand activity at half saturation (X = \). Consider the case of a macromolecule containing N sites that are identical and independent. This is a reference system whose properties will be of great importance in our discussion. The reference system is a scaled replica of the simpler case JV = 1 and as such the relevant thermodynamic quantities X and B are expected to show no substantial difference other than scaling factors. There are essentially two ways of deriving the properties of the reference system. One is simple and intuitive. Since the N sites are identical and independent, the potential II = - k B T i n W of the macromolecule is N times that of a single site. It follows from (1.70) that W(x) = (1 + Kx)N
din* dlnx B = -«*-= dlnx
=
(2.9)
JVK^ 1 + Kx NKx
(1 + Kx)2
(2.11)
The functions X and B scale with N, as expected. The alternative way of deriving the partition function (2.9) is more instructive. Consider the polynomial expansion for the system containing N sites
W(x) = 2 V
(2.12)
The y'th mechano-thermal partition function in eq (2.12) contains the Gibbs free energy of the yth ligated intermediate. Since the sites are all identical and independent, the Gibbs free energy level G; can be written in the form Gj = jG
(2.13)
where G is the Gibbs free energy level of a ligated site and the unligated configuration is assumed to have a free energy level of zero. The degeneracy, or microcanonical component, associated with each Y ; is computed from the number of distinct ways y sites out of TV can be ligated, since any of these possible dispositions is linked to the same value of G ;. The number of such dispositions is given by the binomial coefficient
Cj,N = N\/j\(N-j)\. Hence, Y; = q * e x p ( - Gj/kBT) = q * e x p ( - jG/kBT) = CjtNK! (2.14)
50
2 Global binding processes
where K has already been defined in the simpler case N = 1. Substitution of (2.14) into (2.12) yields ( which is identical to Eq (2.9). The ;th mechano-thermal partition function Y7- in the expansion of *P is related to the Gibbs free energy for the formation of the y'th ligated intermediate from the reference unligated form of the macromolecule, or the reaction M + /XOMX,-
(2.16)
The standard free energy change associated with this reaction is AGj = -kBTlnChN
- jkBTlnK
(2.17)
The degeneracy, CjfN, accounts for the different dispositions of / ligands among TV sites. This term contributes to the free energy change in an additive manner and must be taken into account when dealing with binding to multiple sites (N > 1). The probability of finding the y'th ligated configuration in the reference system is derived from (1.78) as
(1 + Kx)
N
h
N
(
X)N~i = CUNO>(1 - d)N~i
(2.18)
where 6 = X/N is the fractional saturation, or the probability of finding a ligated site in the system. The values of ty as a function of 6 are given in Figure 2.1 for the case N = 4. The functions % and \pN are monotonic, with ^o decreasing and ipN increasing as a function of X or 6. On the other hand, the functions relative to the intermediates of ligation such that 0 < j < N show a distinct peak occurring at intermediate values of saturation. It follows from eq (2.18) that the peak for the function t//; occurs at X = ; , or d = j/N. The yth ligated intermediate has a maximum of occurrence when the system contains j ligated sites on the average. The two limiting configurations with 0 and TV ligated sites are no exceptions to this rule, since they are peaked at X = 0 and X = N respectively. The functions ifi contain all information necessary to define the key properties of the system. In fact, the partition function can be rewritten as V(x) = lCjfNKW = (1 + Kxy^Vj
= Vo1
(2.19)
and equals the inverse of the probability of finding the unligated form of the macromolecule. This result is valid in general, regardless of the
2.1 The reference system
51
00
d
CO
d
CN
d q d 0.0
0.2
0.4
0.6
0.8
1.0
e
Figure 2.1 Plot of the probability of occurrence of ligation intermediates as a function of fractional saturation for a reference system. Curves were drawn according to eq (2.18) with N = 4. The ;th ligated intermediate, as indicated, shows a maximum in the plot for a value of X = /, or 0 = j/N.
specific form of ^o (Hill, 1960, 1984; d'A. Heck, 1971; Schellman, 1975; Wyman and Gill, 1990). Likewise, the functions X and B can be expressed in terms of the ips using eqs (1.133) and (1.134) as follows
X = 2/ty
(2.20)
7=0 N
/ N
\
y=0
\ / =0
/
(2.21) Hence, for a reference system, we have the following important relationship between X and B (2.22)
2 Global binding processes
52
The suffix denotes the particular value of B in a reference system, which will be used in our subsequent analysis. The binding capacity is plotted versus X in Figure 2.2 for the case N = 4. The fluctuations of ligated intermediates around the average ligation value X are reflected by B o. It is a unimodal function of X and goes to zero for X = 0 and X = N, as expected. For X-* 0 the binding capacity tends to vanish due to the low concentration of ligand molecules, while for X-* N all sites become bound and B tends to vanish as no sites are available for ligation. Under these limiting conditions the possible configurations of the system 'freeze' around the average value of X and no fluctuations are possible. The binding capacity is a maximum for X = N/2, where the dispersion of ligated intermediates around the average value is a maximum. The distribution of intermediates in Figure 2.1 provides the physical basis for the behavior of B. The intermediates are all significantly populated as X
6
q 6 0.0
1.0
2.0
3.0
4.0
Figure 2.2 Plot of the binding capacity versus the number of ligated sites for a reference system. The curve was drawn according to eq (2.22) with N = 4.
2.1 The reference system
53
approaches its mid-point value N/2, while only one configuration is present at the limiting values of X = 0 and X = N. The maximum value of B for a reference system is N/4 and provides a measure of the maximum dispersion of ligated intermediates around X. Another important property of the reference system is derived from consideration of the 'affinity function' XK. This function gives the average binding affinity of a site as a function of the ligand activity. If an 'average' site in the system is considered, then the binding reaction site • free + ligand <=> site • bound (2.23) can be associated with an apparent equilibrium constant xK = [site • bound] [site-free] [ligand] The concentration of bound sites is Xmu where mt denotes the total concentration of the macromolecule. Likewise, the concentration of free sites is (N — X)mt and therefore X
K=
(2.25)
(N - X)x This is the equilibrium constant for the reaction (2.23) and gives the affinity function of the system as a function of the ligand activity. The importance of XK will become evident when dealing with cooperative systems. The free energy of binding associated with XK is X
AG = -kBT In
(2.26)
(N - X)x
The affinity function of a reference system can be computed from the definition (2.25) using the expression for X given by (2.10) to obtain X K=K (2.27) X The fact that K does not change with x for a reference system is intuitively obvious. A particularly useful plot is the logarithm of XK versus the logarithm of x. This 'affinity plot' transforms the data shown in Figure 2.3 for Na+ binding to thrombin into those shown in Figure 2.4 (Wells and Di Cera, 1992; Ayala and Di Cera, 1994). The affinity function can be constructed directly from experimental data once X is known as a function of x. For N = 1, or in the case of a reference system for arbitrary N, a plot of lnxK versus lnx should yield a straight line with zero slope. The intercept in this plot gives In K, from which the value of K is determined directly. A familiar transformation of the function X is the Hill plot (Hill, 1910)
2 Global binding processes
54
CO
d
d
CN
d
q d -3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
log x +
Figure 2.3 Plot of X versus In* for Na binding to thrombin, under conditions of: 5 mM Tris, 0.1% PEG, I = 0.2 M, pH 8.0 at 25 °C (Ayala and Di Cera, 1994). The binding reaction involves a single site (Dang, Vindigni and Di Cera, 1995) and obeys eq (2.7) for a reference system, with K = 35 M" 1.
where the logarithm of the ratio X/{N - X), or equivalently 0/(1 - 0), is plotted versus the logarithm of the ligand activity. In the case of the reference system, the Hill plot is a straight line with unit slope, as shown in Figure 2.5. The value of the abscissa where the ordinate vanishes is —In K. The slope of the Hill plot defines the Hill coefficient nn. For a reference system nH = 1, independent of x. The Hill plot is a representation of the function X ln= \nxK + lnx (2.28) N- X and is merely a plot of lnxK rotated by 45° counterclockwise. Hence, the Hill coefficient is the slope of the Hill plot, or the slope of the affinity plot plus one. Other transformations of the function X have little thermody-
2.1 The reference system
55
q CN
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
log +
Figure 2.4 Affinity plot for Na binding to xthrombin. The data are the same as those shown in Figure 2.3, but plotted as h\ K versus lnx. The affinity plot is a straight line with zero slope, as implied by eq (2.27) for a reference system, and yields the value of In K as the intercept of the In* K axis, with K = 35 M"1. namic significance, but are worth mentioning for historical reasons. In the popular Scatchard plot (Scatchard, 1949) given in Figure 2.6, the ratio X/x is plotted versus X. In the case of a reference system the slope in this plot is equal to —K and the intercept on the ordinate is NK, as can be easily verified from eq (2.10). In the equally popular double-reciprocal plot given in Figure 2.7 the function 1/X is plotted versus 1/x. The slope in this plot equals l/NK and the intercept is equal to 1/iV. The Scatchard and double-reciprocal transformations are mentioned solely for the sake of completeness and will not be pursued any further in our treatment. These linear transformations were invaluable in those days when nonlinear least-squares analysis of experimental data could not be performed routinely on a computer. These plots, however, convey very little information in thermodynamic terms and should be used with extreme caution in
2 Global binding processes
56
i £
^
o CO
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
log x +
Figure 2.5 Hill plot for Na binding to thrombin. The data are the same as those shown in Figure 2.3, but plotted as ln[X/(l - X)] versus lnx. The Hill plot is a straight line with a slope of one, as implied by eq (2.28) for a reference system. The intercept on the abscissa where the ordinate vanishes gives the value of 1
practical applications. The Scatchard plot may grossly underestimate the number of binding sites (Klotz, 1985), while the double-reciprocal plot may yield severely biased estimates of binding parameters (Di Cera, 1992a). These pitfalls, which were very well known to Scatchard (1949), continue to be ignored by many experimentalists in their data analyses. As a general rule, data should be plotted and analyzed in the form taken experimentally, with a distribution of errors in the weighting scheme for parameter estimation that reflects the one derived from actual measurements. The problem of correct parameter estimation in data analysis is of considerable importance and complexity. The interested reader is referred to the excellent monographs by Bard (1974) and by Box and Jenkins (1976) for a detailed treatment.
2.2 Cooperative binding
57
q d
q d
o
CM
q d
q d 0.0
0.2
0.4
0.6
0.8
1.0
X +
Figure 2.6 Scatchard plot for Na binding to thrombin. The data are the same as those shown in Figure 2.3, but plotted as X/x versus X. The plot is linear, with an intercept on the ordinate of K = 35 M" 1. The error of data points at low saturation is emphasized in the Scatchard plot and complicates the resolution of binding parameters (Klotz, 1985). 2.2 Cooperative binding
In the reference system all sites are alike and independent. The partition function assumes a simple binomial form raised to the Nth power, where N equals the number of sites in the system. In general, one can expect the sites to be different and/or influence each other's binding reactions. In this case, of considerable interest in biology, the simple rules examined for the reference system no longer apply. The partition function of a macromolecule containing N sites is N
(2.29)
7=0
The yth mechano-thermal partition function in eq (2.29) contains the
2 Global binding processes
58 o
CO
o
o
CO
q o
q d 0.0
300.0
600.0
900.0
1200.0
1500.0
1 /X Figure 2.7 Double-reciprocal plot for Na+ binding to thrombin. The data are the same as those shown in Figure 2.3, but plotted as l/X versus 1/x. The plot is linear with a slope equal to 1/K, with K = 35 M" 1. Note how the data for x small significantly deviate from the straight line, unlike those obtained for large x. This is a consequence of the serious bias introduced by the transformation (Di Cera, 1992a).
Gibbs free energy of the y'th ligated intermediate. Since there are N + 1 such intermediates, and one can be taken as reference, there are N independent mechano-thermal partition functions in eq (2.29). This is the fundamental difference with the reference system that can be fully described in terms of the free energy of binding to any of the TV equivalent and independent sites. Consider the reaction M + jX e> MXy (2.30) where the yth ligated intermediate is generated from the unligated form. The standard free energy change associated with this reaction is (d'A. Heck, 1971)
2.2 Cooperative binding
59
}. = kBTln— = -k BTlnAj
(2.31)
where Aj is the yth overall equilibrium binding constant. There are N such independent constants that define the partition function of the system. The degeneracy associated with each Y ; is again given by CjyN, or by the number of dispositions of y ligands among N sites. Hence, (2.32)
and Go = 0 can be assumed without loss of generality. If the sites are all alike and independent, as in the case of the reference system, the free energy level G7 is just jG, where G is the same as the free energy change for binding to any of the sites. The partition function in this case factors out in terms of N identical binomials. If the conditions required by the reference system are not met, no such simplification is possible for G7 and N independent binding free energies must be considered. Also, the partition function does not factor out in terms of identical binomials. This is the mark of cooperative binding phenomena. In order to understand the onset of cooperativity better, it is useful to define a new set of equilibrium constants, the stepwise constants, k, that reflect the binding affinity of sequential ligation steps. Unlike the As, the ks have the same dimension (M"1) and can be compared directly. Consider the free energy level G7 in eq (2.32) split into y terms as follows Gj = G[ + G'2 + . . . + G; (2.33) Since G7 is a state function defining the free energy level of the yth ligated intermediate with respect to the reference, unligated configuration, it makes no difference how many intermediate free energy terms we use to define it. Hence, we choose a set of free energy values such that G7 defines the free energy change due to adding one ligand to the (y - l)th ligated configuration to yield the yth ligated one. The equilibrium constant associated with G7 is (2.34)
and defines the stepwise equilibrium binding constant for the yth ligation step. The connection between overall and stepwise constants is Aj = CJ,Nk1k2 . . . kj
k, - <±±!LAL = CjiN
A
h l
I
(2.35)
AL
N - j+1 A
h l
(2.36)
2 Global binding processes
60
where it should be recalled that C0^N = CN}N = 1 and Ao = 1. In the case of the reference system, A}- = CjNKj and kx = k2 = . . . = kN = K. All ligation steps have the same affinity. A cooperative system is characterized by a change in affinity with ligation. If the affinity increases with ligation we have positive cooperativity, otherwise cooperativity is negative. For N ^3 cooperativity can also give rise to mixed patterns, where increases in affinity alternate with decreases and vice versa. For a practical point of view, the definition of positive and negative cooperativity in a macroscopic sense refers explicitly to the comparison of the first and last steps of ligation. Specifically, positive macroscopic cooperativity demands k N> kx, while negative macroscopic cooperativity demands kN< kx. Regardless of the nature of the binding process under investigation, the relevant response functions X, B and XK can be determined in a straightforward manner from the partition function (2.37) i=o
cast in terms of the overall equilibrium constants for the sake of simplicity. The average number of ligated sites is given by the expression
x-tai-e—i,)
(2.38)
dlnx
7=0
and is bounded from zero to N. The binding capacity is given by 2
B = dX /=o din*
7=0
;=0
= (?) - (jy
(2.39)
;=0
/
and is bounded from zero to X(N — X). The affinity function is given by N
X
K =
I
X
7=0
(TV - X)x
(2.40)
i=o X
It is straightforward to show that K reaches two limiting values for x -> 0 and x —> oo respectively. These limits are
2.2 Cooperative binding °K = — = k t N =kN
61 (2.41a) (2.41b)
and are equal to the first and last stepwise binding constants. This is why the stepwise constants are important in defining macroscopic cooperativity. The affinity function for TV = 2 is plotted in Figure 2.8 for a negatively (k2 < kx) and positively (k2 > kx) cooperative system. It is quite instructive to note from eq (2.40) that the affinity function approaches its limiting values with a zero slope. Hence, at either low or high saturation a cooperative system behaves like a reference system, with an affinity K = ki for x —> 0 and K = kN for x —»oo. The physical basis of this effect is as follows. At low saturation, binding to the macromolecule produces ligation of the first site, while at high saturation binding produces ligation of the last site. In either case, the cooperative nature of the system is not reflected in its macroscopic behavior since ligation involves only one site. The properties of XK are of particular relevance in understanding the physical basis of cooperativity. The average number of ligated sites can be written in a rather simple way by using the definition of XK, i.e., X-
"**
1 + xKx
(2.42)
In a reference system XK = K, independent of x9 and X assumes the familiar form (2.10). In general, a cooperative system can be regarded as a reference system with variable affinity. There is always a unique value of X K for a given x and, for this particular value of x, the function X of a cooperative system assumes the same value as the function X of a reference system with K =XK. This is illustrated in Figure 2.9 for both positive and negative cooperativity. In making its transition from °K = kx to °°K = kN, a cooperative system explores a continuous spectrum of reference systems. Also, the function X is the same as X for a reference system with K = kx at low saturation and K = kN at high saturation. When the system makes a transition from low to high affinity, as in the case of positive cooperativity, the function X becomes steeper than any intermediate reference curve crossed over at intermediate saturation values, as shown in Figure 2.9(b). Actual experimental data are shown in Figure 2.10(6) for carbon monoxide binding to hemoglobin (Perrella et ai, 1990). The slope in the lnxK plot is positive at intermediate saturation, as shown in Figure 2.8(fc), and the binding capacity is higher and
2 Global binding processes
62
(a)
o 6
-3.0
-2.0
-1.0
0.0
1.0
2.0
1.0
2.0
3.0
log x
(ft)
*
o W> 6
o
-3.0
-2.0
-1.0
0.0
3.0
log x Figure 2.8 Affinity plot for N = 2 in the case of (a) negative or (b) positive cooperativity. Curves were drawn according to eq (2.40) with (a) k\ = 10 M" 1, jt2 = o.l M"1, and (b) kx = 0.1 M"1, k2 = 10 M"1. Chain lines depict the values of the In ki and In k2 asymptotes in the plot (see eqs (2.41a) and (2.41b)), and the value of ln*^ at half saturation, i.e., ln\/ kik2. These values correspond to the value of XK for reference systems with K = fcl5 K = k2 or K = y k\k2.
2.2 Cooperative binding
-3.0
63
2.0
3.0
Figure 2.9 Binding curve for N = 2 in the case of (a) negative or (b) positive cooperativity. Curves were drawn according to eq (2.38) with (a) kx = 10 M"1, k2 = 0.1 M"1, and (b) kx = 0.1 M " \ k2 = 10 M"1. Alternatively, the function X can be derived from eq (2.42) using XK as shown in Figure 2.8. Chain lines depict binding curves of reference systems for which K = k\, K — \J k\k2, or K = k2.
64
2 Global binding processes s
(a)
1
0 /
1
1
Jv '
1 1
1 1
VJ
1
i
1
' '
rfT
1
1
i
JRj
1
1 1
X S
Cy i
I
1
X.'
r#
• i i
1
i
1 1
Q>' )r
I
i
'
Go l
i i
1 of
1
i i
,'
i i
1
t
1
r /
i i
i
I
/ q d -10.0
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log [dT(pT)15] (M)
q
CO
q d -3.0
-2.5
-2.0
-1.5
-1.0
logpCO (Torr)
-0.5
0.0
2.2 Cooperative binding
65
more narrowly peaked, as shown in Figure 2.11(fc). On the other hand, when the system makes a transition from high to low affinity, as in the case of negative cooperativity, the function X becomes less steep than any intermediate reference curve crossed over at intermediate saturation values, as shown in Figure 2.9{a) and in the case of dT(pT)15 binding to E. coli single strand binding protein (Bujalowski and Lohman, 1989) depicted in Figure 2.10(a). The slope in the \nxK plot is negative at intermediate saturation, as shown in Figure 2.8(
A landmark feature of positive cooperativity is the significant suppression of intermediates, with the shape of the binding curve, or any other response function, being determined mostly by the unligated and fullyligated species. The relative insensitivity of positively cooperative binding isotherms to intermediates of ligation makes resolution of binding constants for such systems a rather nontrivial task. This also complicates significantly the formulation of specific molecular mechanisms of interaction. Figure 2.10 (a) Binding curve of dT(pT)i5 to E. coli SSB tetramer under conditions of: 10 mM Tris, 0.1 mM Na3EDTA, 50 mM NaCl, pH 8.1 at 25 °C (Bujalowski and Lohman, 1989). The continuous line was drawn according to eq (2.38) with the four stepwise constants: kx = 3.4 x 107 M"1, k2 = 7.3 x 106 M"1, k3 = 8.9 x 104 M"1, k4 = 2.8 x 104 M"1. The values of kx and k4 were used to construct the binding curves for the reference systems, given by chain lines (K = k\ for the curve at left, K = k4 for the curve at right). The reference curve (b) Carbon monoxide binding in the middle corresponds to K = {k^k^k^)1^. curve of hemoglobin under conditions of: 3.125 mM heme, 0.1 M KC1, pH 7.0 at 20 °C (Perrella et al., 1990). The continuous line was drawn according to eq (2.38) with the four stepwise constants: k\ = 2.5 Torr" 1, k2 = 8.5 Torr"1, k3 = 225 Torr"1, k4 = 1460 Torr"1. The values of k\ and k4 were used to construct the binding curves for the reference systems, given by the chain lines (K = k\ for the curve at right, K = k4 for the curve at left). The reference curve in the middle Note the different scale on the abscissa in the corresponds to K = {k^k^k^4. two cases. The binding curve for the negatively cooperative case (a) spans nearly eight orders of magnitude in the ligand activity, as opposed to only two for the positively cooperative case (b).
2 Global binding processes
66
CO
6
q d
-3.0
oo
d
CO
d
o d -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x
Figure 2.11 Binding capacity for N = 2 in the case of (a) negative or (b) positive cooperativity. Curves were drawn according to eq (2.39) with (a) kx = 10 M"1, k2 = 0.1 M"1, and (b) kx = 0.1 M"1, k2 = 10 M"1. Chain lines depict the binding capacity of reference systems for which K = k\, K = V kik2, or K = k2.
2.2 Cooperative binding
-7.0
-6.0
-5.0
-4.0
67
-3.0
log [dT(pT)15] (M)
-3.0
logpCO (Torr)
Figure 2.12 Plot of the probability of occurrence of ligation intermediates as a function of the logarithm of ligand activity for (a) the binding of dT(pT)i5 to E. coli SSB tetramer (Bujalowski and Lohman, 1989), and (b) carbon monoxide binding to hemoglobin (Perrella et ah, 1990). The continuous curves were drawn according to eq (2.43), with parameter values given in the legend to Figure 2.10. The distribution of intermediates should be compared with that of a reference system for N = 4 given in Figure 2.1. In the negatively cooperative case (a), intermediates make a larger contribution at intermediate saturation, while in the positively cooperative case (b) the contribution is almost negligible.
68
2 Global binding processes
The slope of the affinity plot is related to a classical measure of cooperativity. The Hill coefficient (Hill, 1910; Cantor and Schimmel, 1980; Wyman and Gill, 1990) din
nH =
\N-Xj
= ""* + 1 = AX + 1 (2.44) d In x d In x is the slope of the Hill plot, which is the affinity plot rotated 45° counterclockwise. The value of nu is equal to the value of the slope of the affinity plot, AX = d l n ^ / d l n x , plus one. The significance of AX as an important linkage property of the system will become evident in Chapter 3. The properties of the Hill coefficient can be derived from those of the affinity function in a straightforward manner. The Hill plot approaches asymptotically two straight lines of unit slope (Wyman, 1964), In kx + lnx for x —> 0 and In kN + lnx for x —> oo, as shown in Figure 2.13. Positive cooperativity in the Hill plot corresponds to nu > 1 and negative cooperativity to nH < 1. The Hill coefficient has nothing to do with 'the number of interacting sites in the system', a rather misleading definition which is often used. It represents a measure of the change of X with lnx and, as such, it is related to the binding capacity. It is this connection that makes the Hill coefficient of thermodynamic interest in the discussion of binding phenomena. From the definition of nH in eq (2.44) it follows that (Schellman, 1990a) NB B nu = ™ = -E(2.45) X(N-X) Bo The Hill coefficient is the ratio between the binding capacity of the system and that of a reference system with the same number of sites. As such, it represents the ratio of the fluctuations of the ligation intermediates around the average value defined by the binding isotherm, relative to that of the reference system. A value of nu = 3, as found for oxygen binding to the four sites of hemoglobin (Chu, Turner and Ackers, 1982), means that the binding capacity of hemoglobin is three times that of a reference system of four sites. A more interesting interpretation of the Hill coefficient will be given in Chapter 3 when we consider site-specific binding effects. 2.3 Properties of the binding curve
We now examine the properties of the binding curve X. Wyman was the first to recognize a fundamental property of X that can be exploited from analysis of experimental data (Wyman, 1964). If measurements of X are
2.3 Properties of the binding curve
69
§ o
-3.0
-3.0
2.0
-2.0
-2.0
-1.0
3.0
0.0
log x Figure 2.13 Hill plot for N = 2 in the case of (a) negative or (b) positive cooperativity. Curves were drawn using parameter values (a) ki = 10 M" 1, k2 = 0.1 M"1, and (b) kx = 0.1 NT1, k2 = 10 M"1. Chain lines depict the Hill plot of reference systems for which K = k\, K = V k\k2, or K = k2.
70
2 Global binding processes
available, it is always possible to obtain the coefficient AN of the partition function by direct integration. The Wyman integral equation
lnx m = — fN\nxdX
(2.46)
NJo defines the mean value of lnx. The value of xm associated with eq (2.46) is hereby defined as the 'mean' ligand activity 1 to indicate the value of x for which lnx assumes its mean value. Alternatively, the definition (2.46) can be interpreted as the mean value of the ligand chemical potential in k B T units, or the average work spent on ligating one site of the system. The importance of the quantity xm stems from its connection with the last coefficient of the partition function, AN. The mean ligand activity is, in fact, the value of x where the unligated and fully ligated configurations of the macromolecule are equally populated. This follows directly from the definition of AN and solution of the integral equation (2.46). Integration by parts yields lr00 1 Wioo) l v 7 X d l n x = ln(oo) - —In = - — lnAN lnx m = ln(oo) - — NJ-oo N W(0) N (2.47) But from the law of mass action one has
A^JMM
(2 .48)
Hence, for x = xm9 the concentration of the fully ligated species equals that of the reference, unligated configuration. The value of xm uniquely defines the last coefficient of the partition function and, vice versa, A N uniquely defines the value of the mean ligand activity. The connection with the total work of ligating the macromolecule should be emphasized. This quantity is expressed by the standard free energy change AGN=
-kBT
In AN
(2.49)
as implied by eq (2.31). The average work spent in ligating one site of the macromolecule is the value of AGN divided by N, or the logarithm of xm in kBT units. This important property of the system can be derived from analysis of X in a direct way, without knowing the detailed form of the partition function. The mean ligand activity should not be confused with Xx/2, which gives the value of x at half saturation where X = N/29 although it is identical to this quantity in some special cases. The importWe employ for xm the term 'mean' in place of 'median' ligand activity, as originally defined by Wyman (1964), to be consistent with the mathematical definition in eq (2.46).
2.3 Properties of the binding curve
71
ance of xm in the analysis of linkage effects involving multiple ligands will be illustrated in Section 2.6. In practice, calculation of xm from measurements of X is extremely simple. Wyman pointed out that eq (2.46) can be rewritten as N
1 r x ±f In —dX =
NJo
(2.50)
xm
whereby integration by parts yields r\nxm
j
r
|
(2.51)
(N -
The value of In xm defines the value of In x where the area under the curve X, <2U, at the left-hand side of eq (2.51), equals the area above the curve X up to the asymptote X = N, Qa, as shown in Figure 2.14. Wyman's equal-area rule is an elegant property of X that is widely used in practical
/
*^
q d -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x Figure 2.14 Binding curve of a two-site macromolecule. Qu is the area under the curve, from — oo up to the point considered, while Qa is the area above the curve up the asymptote X = N, from the point considered to + oo.
72
2 Global binding processes
applications. There is, however, an important property of xm that should be emphasized for its operational significance (Di Cera, 1994a). Let us define the functions Qu and Qa as follows QU(X) = f ln *Xdln*'
(2.52a)
J — 00
Qa(jc) = C (N - X)dlnxf hnx
(2.52b)
It follows immediately from eq (2.51) that (2.53) lnx m = lnx + —[Q a(x) - Qu(x)] TV for any value of x. Therefore, lnx m is an invariant property of the binding curve X, since for any two points xx and x2 one has In*! + —[G a(xi) - Qu(*i)] = lnx 2 + — [Q a(x2) - Qu(x2)] = \nxm N N (2.54) As a result of this invariant property, the mean ligand activity can be measured from any point along X. Given the arbitrary point lnx1? the value of Qu(xi) is given by the area under the curve X from — oo to lnxx and likewise the value of <2a(*i) is given by the area above X up to the asymptote X = N, from lnjci to oo. The difference Qa(*i) ~ Qu(*i)> divided by N and added to liucl5 gives lnx m , regardless of the point xx. Wyman's equal-area rule is embodied by the invariant property (2.53) as a special case. Another property of the binding curve X arises in connection with the analysis of its shape. The binding curve X plotted versus \nx is symmetric when it can be reproduced exactly upon a rotation of 180° around a point of coordinates lnx s , X = JV/2 (Wyman, 1948; Whitehead, 1980a; Weber, 1982; Di Cera, Hopfner and Wyman, 1992). The point xs is the center of symmetry of X. Symmetry introduces mathematical constraints among the otherwise independent coefficients of the partition function *¥. From a mathematical point of view, a function F(x), with -oo ^ x ^ oo? can show a number of symmetry properties. If F is an even function, then F(x) = F(—x) and any point in the positive half plane at x has an image at —x. Symmetry in this case is specular since F in one half plane is the specular image of F in the other half plane. If F is an odd function, then F(x) = —F(—x) and symmetry is rotational since F is reproduced exactly when rotated 180° around x = 0. Specular and rotational symmetries are related by differentiation. It is easy to prove that if F has rotational
2.3 Properties of the binding curve
73
symmetry, then its derivative <\F/dx must have specular symmetry. Hence, if X is symmetric around lnx s , then the binding capacity B must have specular symmetry around lnx s . The condition of rotational symmetry for a generic function F(x), where — oo ^\nx ^ o° and xs is the value of x at the center of symmetry, is given by F(xsk) + FOsA"1) = F(0) + F(oo)
(2.55)
for any A ^ 0. At the center of symmetry the value of F is by definition halfway between the limiting values F(0) and F(oo), i.e., F (,,)
= f<°>+ f<°°> 2
(2.56)
so that F(xsX) + F^A" 1 ) = 2F(xs)
(2.57)
Using the function X in eq (2.57), we obtain the rather simple relationship X(xsX) + X{xsX~l) = N
(2.58)
since X(xs) = N/2. It follows immediately from eq (2.58) that if X is symmetric, then in a plot of X versus lnx the values of X computed at points equidistant from the center of symmetry always add up to the total number of sites N. This situation is illustrated in Figure 2.15. Let 5= |lnA| be the distance of a point from the center of symmetry lnx s along the lnx axis. It follows from the condition (2.58) that the area under the curve X from In JCS — 5 to In xs must equal the area above the curve X, up to the asymptote X — N9 from lnx s to lnx s + 8. This is because X is reproduced exactly upon a rotation of 180° around its center of symmetry. In mathematical terms one has rlnA;s
I
J\nxs-d
r\nxs+d
Xdlnx = I
Anxs
(N - X)dlnx
(2.59)
or r\nxs+d
X d l n * = Nd
(2.60)
\nxs—d
The area under the curve X within two points symmetrically disposed at a distance 6 from the center of symmetry equals the area of a rectangle of base 6 and height N (see Figure 2.15). This condition holds regardless of the value of 8 in the range from zero to oo. For a symmetric binding curve the center of symmetry necessarily coincides with the value of x at half saturation, i.e., xs = x1/2. We have already pointed out that the mean ligand activity xm does not always coincide with x^. This is illustrated by a
2 Global binding processes
74 q
q d
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x Figure 2.15 Symmetric binding curve of a two-site macromolecule. The two points shown are equidistant from the center of symmetry, lnx s = 0, and their X values sum up to two, as implied by eq (2.58). The area under the curve, from the lower point to the center of symmetry (vertical shades), is equal to the area above the curve, from the center of symmetry to the upper point (vertical dashed shades). This holds for any pair of points at a distance 6 from the center of symmetry, as implied by eq (2.59). It also follows from eq (2.60) that the area under the curve between any two points symmetrically disposed at a distance <5 from the center of symmetry (vertical shades and crossed shades) equals the area of a rectangle of base 6 and height TV (vertical dashed shades and crossed shades).
transformation of eq (2.60). Integration by parts yields N/2+a N/2-a
x
In —
(2.61)
Xs
Here 0 ^ a ^ N/2 represents the difference between the value of X at a distance <5 from the center of symmetry and half saturation. The similarity between eq (2.61) and eq (2.50) is evident. However, eq (2.50) holds only for a = N/2, while eq (2.61) is valid for any a in the range from zero to
2.3 Properties of the binding curve
75
N/2. Hence, if X is symmetric, then xs — xm = xXj2 follows necessarily and the mean ligand activity coincides with the value of x at half saturation. The reverse is not necessarily true, since an asymmetric binding curve can be such that xm = xx/2. On the other hand, if xm i= xx/2, the binding curve is necessarily asymmetric. In conclusion, the value of Xi/2 should never be used as an estimate of x m , although xm = x1/2 holds in the case of symmetric binding curves and other special cases. Another interesting consequence of the symmetry of X is the symmetry of the affinity function. If X is symmetric, then lnxK is itself symmetric and has the same center of symmetry (Di Cera et«/., 1992). In fact, from eq (2.58) it follows that X(xsX) = N - X{x,X~l)
(2.62a)
X{x,k~l)
(2.62b)
= N - X(xsX)
Upon multiplying eqs (2.62a) and (2.62b) together, rearranging and taking the logarithm of both sides, one has lnx*xK 4- lnx^K
= -21nxs
(2.63)
X
Application of eq (2.57) and the definition of K in eq (2.25) show that if X is symmetric, then lnxK is also symmetric and has the same center of symmetry. The term at the right-hand side of eq (2.63) is in fact twice the value of lnxK computed at its center of symmetry, which coincides with that of the function X. Symmetry imposes constraints on the coefficients of the partition function. Some qualitative, although important, conclusions can be derived intuitively. Since a symmetric binding curve can be reproduced exactly by rotation around the center of symmetry, it follows that knowledge of half of the binding curve (say X from zero to N/2) completely defines the other half. Accordingly, only half of the coefficients of the partition function are truly independent. In fact, integration of eq (2.60) yields
W(xsX) = A ^ O a " 1 )
(2.64)
which implies that the partition function W is a symmetric polynomial when x is expressed in xs units. This is demonstrated by rewriting eq (2.64) as follows ^(Ajxl - AN.fc^)ki
=0
(2.65)
Since eq (2.65) must hold for any A, the coefficients of each power of A necessarily vanish. The term with j — 0 leads to xs = A^I'N\ and hence
76
2 Global binding processes
xs = xm as already established. Elimination of xs from all the other terms in eq (2.65) yields the condition of symmetry for the equilibrium constants as follows (Wyman, 1948) Afa'/N = AH-^-M
(2.66)
which holds for any j if X is symmetric. The condition above states in more explicit form the fact that the partition function W is itself symmetric when x is expressed in xs or xm units. The change of variable y = x/xm in W yields a transformation whereby any value along the In x axis is shifted by a constant amount lnx m . The transformation is equivalent to a 'normalization' and does not affect the shape of any of the response functions X, B or XK. The normalized form of the partition function is W(y) = 1 + AlXmy
+ A2x2my2 + . . . +
ANx^yN
= 1 + axy + a2y2 + . . . + yN
(2.67)
where
Symmetry implies that the coefficient of the yth power of y in V is the same as that of the (N — y)th power of y9 as seen for symmetric normalized polynomials. As a result, knowledge of half of the equilibrium constants completely defines the partition function. For N = 1 or N = 2 the condition (2.66) is a mere tautology. For N = 3 the condition is A3 = (A2/A^f or kik3 = k\. For N = 4 the condition of symmetry is A4 = (Ai/Ax)2, or kxkA = k2k3 and does not depend on A2. In general, the condition of symmetry involves all As in the partition function for N odd and all As but AN/2 for N even. 2.4 Properties of the binding capacity In Section 1.3 we established that the binding capacity parallels physical response functions such as the heat capacity and the compressibility and is strictly related to the stability principles embodied by the second law of thermodynamics. The function B is always positive and an increase/ decrease in the chemical potential of X always leads to an increase/ decrease of X. From a mathematical point of view, the positivity of B implies that X is a monotonic function of lnx whose derivative never vanishes for finite values of x. In other words, the equation B = 0 has no solutions for finite x. This important property of B can be proved mathematically in a direct way (Di Cera, 1992b). Let us rewrite eq (2.39)
2.4 Properties of the binding capacity
11
as follows N 7=0
B =—
N
/
7=0
'—
^
N
V=0
\
_
/
(2.69)
The denominator of eq (2.69) is always positive, since the partition function has all positive coefficients and cannot vanish for any x ^ 0. The sign of B depends solely on the expression in the numerator, which can be rewritten as N N i=0;=0
Since it makes no difference in which order the double summation is carried out, interchanging the indices i and j yields AT N
7=0/=0
Hence,
- 7)2AV+/
P(*) = lH^
(2.72)
2/=o/=o and p is a polynomial with all positive coefficients and cannot vanish for any finite value of x. The only two values of x for which B = 0 are x = 0, since p(0) = 0, and x = °°, since the denominator of (2.69) is a polynomial of higher degree in x than p itself. Another property of B is of considerable interest. Measurements of X yield information on xm and hence the last coefficient of the partition function AN9 through the integral equation (2.46). Information on the other coefficients of the partition function cannot be derived by numerical integration of X and requires nonlinear least-squares analysis of the data. However, there is another set of relationships such as (2.46) which allows us, at least in principle, to determine uniquely the coefficients of W from numerical integration of X and B. Let us rewrite eq (2.46) as follows, using the definition of B lnxm = — f
lnxtfdlnx
(2.73)
NJ-00
A close examination of eq (2.73) reveals an important aspect of B, first pointed out by Sturgill and Biltonen (1976). The integral (2.73) is equivalent to the calculation of the average value of In x over the entire range of
78
2 Global binding processes
definition of this variable. The weighting function for the integration is the binding capacity B divided by N. The normalization condition Bdlnx = TV
(2.74)
0
makes this property of B even more evident. The area under the B curve is always finite and equals the total number of sites, independent of the particular form of B. The function B/N can thus be regarded as the probability density function of the variable lnx (Di Cera and Chen, 1993). Calculation of lnx m is equivalent to calculation of the first moment of B/N. Since AGm = kBT In xm is the average work spent in ligating one site of the macromolecule, the ratio B/N computed at x = § and multiplied by dln§ gives the probability fraction, dv, that AGm has a value lying between kBTlnt; and kBT(ln% + ding) (see Figure 2.16(6)). Alternatively, this probability fraction can be derived from the plot of X versus lnx by computing the slope at x = £ and dividing the result by the number of sites N9 as shown in Figure 2.16(a). The most probable value of AGm is derived from the first moment of the B/N distribution. We now consider higher amounts of B/N, i.e.,
(In**) = — f° \nkxBdlnx
(2.75)
NJ-OO
The connection with the coefficients of the partition function is obtained from the analytical solution of eq (2.75). Sturgill and Biltonen (1976) have provided the solution for the first two moments in a number of cases of interest. Here we give the solution in the general case, using the moment generating function (Di Cera and Chen, 1993)
G((o) = fx^Bdx
= fx^dX
(2.76)
Jo Jo The function G can be cast in terms of B or X, as indicated by eq (2.76), and as such it can be constructed directly from experimental data collected either as B or X measurements. However, for the purpose of obtaining an analytical solution for G we will use the properties of B which, unlike X, is related to a probability density function. From the definition (2.76), the kth moment of the B/N distribution is derived as (Feller, 1950) N
Qk
(2.77)
The generating function G exists for 0 < co < 1, by virtue of the normaliza tion condition (2.74).
2.4 Properties of the binding capacity
79
CM
(a)
0.5
H
1.0
ID
o -1.0
0.0
3.0
log x
-1.0
log x
Figure 2.16 Binding capacity as a probability density function. The value of B/N at x = £, times dln£, gives the probability fraction, dv, that AGm lies between kBTIn % and kBT(In % + dln£). This is equivalent to computing the probability density that the logarithm of the mean ligand activity lies between ln§ and In § + d In g. The value of B/N can be derived either from the curve X as the slope at x = | divided by iV (a), or directly from a plot of B/N (b). In the example shown one has B/N = 0.2 at x = £ (log § = 0.4), which implies that dv/dln§ = 0.2. The probability density that log;cm lies between 0.4 and 0.4 + Alog§is0.2Alog!.
80
2 Global binding processes
We now rewrite the partition function W in the factorial form W(x) = (1 + <xxx){l + a2x) . . . (1 + aNx) (2.78) where a}- is the yth coefficient of the factorization related to the yth root of *P. Klotz (1985, 1993) refers to these coefficients as binding constants of 'ghost' sites. Since W is a polynomial of degree N with real and positive coefficients, the partition function is analytical everywhere on the positive real axis, which is the one of physical significance. The coefficients a can be real and positive, or complex conjugate. The connection between the nature of the coefficients a and cooperativity will soon appear evident. Since W can be cast in a simple form such as (2.78), regardless of the nature of its roots, one has X =^
~J~
(2.79)
(2.80) ~i (1 + o-x)2 Hence, the solution of eq (2.76) is itself a linear combination of N similar terms of the form
8(a)= f "*" dx Jo (1 + ax)2
(2.81)
and convergence of the integral necessarily demands 0 < co< 1. If a is real, then the transformation 0 = ax/{\ + ax) yields g((o) = I a~w0(D(l — 0)~CD d0 = a~w (2.82) sin 77 co Jo If a is complex, then g is obtained from the theorem of residues (Wilson, 1911) by properly choosing an integration path in the complex plane. Introducing the complex variable z = ax in eq (2.81) yields
g(co) = f a"°z\
dz
(2.83)
where y denotes integration along the semi-line in the direction of a in the complex plane. Since the function zw/{l + z)2 is holomorphic in the region / f shown in Figure 2.17, the path integral along a closed contour C must vanish by virtue of Cauchy's theorem. Hence, f a~Wz™ dz = f + f + f + f = 0 (2.84) 2 Jc(l + z) J/i J/2 J/3 Ju The integral Ix computed in the complex plane along the semi-line in the
2.4 Properties of the binding capacity
81
Figure 2.17 Integration path in complex plane for the integral (2.84). The argument of the integral is a holomorphic function in if and the path integral vanishes according to Cauchy's theorem.
direction of a for R —> o° and r —> 0 is the required solution for g. The integral I2 is such that dz
(R - I) 2 (2.85)
o (i? - I) 2
and goes to zero for /?-» °o, since a;< 1. The integral /3 involves the positive real axis only. The integral 74 is such that
h=
—
JiJl
CO-CO
+ zf
dz
(• l*-1l*1| dz | ^ f\ar°\r»°
J/4 U + z\2
Jo (1 - r)2
(1 - rf
(2.86) and goes to zero for r —» 0. The solution for g is determined solely by the value of —/ 3, i.e., ayc Jo (1 + ay)
2
dy — a
7T0)
SIMTW
where y = x/ac. The solution is identical to that found in the case of a real and positive. Hence, the general form of G is 7TO)
smna)
_ aja
(2.88)
]=i
To obtain the moments of eq (2.77), we take the Taylor expansion of
82
2 Global binding processes
G = fh, where f(a)) = -^— smirco h(co) = a p + a ^ + . . . + o r ^ The Taylor expansion of / is
/(a,) = 1 +
(2.89a) (2.89b)
i 2 < 2 2 t "- 1 >I^K™>*
(2.90) *=i (2fc)! where B2k is the 2A:th Bernoulli number (Gradshtein and Ryzhik, 1980). The Taylor expansion of h is
,/ x *, £ i £ G^ln*.- £ co3 In3 a/i(ai) = N - 2>lnar,. + 2 — ^ " E — r p ^ + y=i
y=i
7=i
2!
3!
2!
•• • 3!
(2.91)
Hence,
(inx) (1tfx)
=
(2.92a)
N
3
f
(In3 x) = «*>
15
+ 2n2C2
(2.92b)
c3
(2.92c) (2.92d)
and so on. The first moment equals lnx m . The second moment is proportional to the sum of squares of the logarithm of each coefficient a, and in general the A:th moment is expressed as the sum of all even/odd powers of each In a term up to order A:, for A: even/odd. The Cs are uniquely defined in terms of the moments and themselves uniquely define the sum of the various powers of In ar;s. Hence (Di Cera and Chen, 1993), Theorem: The first N moments of the binding capacity uniquely define the N independent coefficients of the partition function. This theorem demonstrates how the statistical properties of B can be exploited for the determination of the coefficients of x¥. The theorem is general and applies to any thermodynamic ensemble whose partition function can be cast in terms of a polynomial of finite order. Binding
2.4 Properties of the binding capacity
83
capacity measurements have been used to resolve the moments of the B/N distribution (Mountcastle, Freire and Biltonen, 1976; Sturgill, 1978; Sturgill, Johnson and Biltonen, 1978; Di Cera and Chen, 1993). These measurements have demonstrated that it is possible to decouple the first two moments of the B/N distribution in a satisfactory way. The first two moments of the B/N distribution are of particular importance in practical applications. The first moment contains information on the average free energy of binding (in kBT units) per site. However, this quantity per se provides no information on cooperativity. Cooperativity is reflected by the steepness of the binding curve and hence by the binding capacity. This situation is illustrated in Figure 2.18 for the case of positive
q CO
q
q 6 -1.0
0.0
1.0
2.0
3.0
logpO2 (Torr) Figure 2.18 Binding capacity for N = 4 for various cooperative cases. Points depict the B distribution for oxygen binding to hemoglobin (Di Cera, Doyle and Gill, 1988b). The dashed curve is the best fit of the data. The continuous curve refers to the reference system, while the chain curve refers to a negatively cooperative system. The three curves have the same value of N and mean ligand activity, but differ substantially in shape.
84
2 Global binding processes
and negative cooperativity. The three curves depict different B distributions with the same value of xm. The difference in cooperativity stems from the dispersion of the distributions rather than the position along the In A: axis, which defines lnx m . The dispersion is related to the variance o2 = (In 2 *) - (lnx) 2
(2.93)
The variance of B/N is related to the cooperativity of the system. This is intuitively obvious because the area under the B curve is finite and if B increases and becomes more sharply peaked, then o2 must decrease to keep the area constant and vice versa. We know from eq (2.45) that the ratio of the binding capacity of the system with respect to that of a reference system containing the same number of sites gives the Hill coefficient, i.e., a classical measure of cooperativity. The more peaked J5, the higher the cooperativity. Hence, we expect the smaller the variance, the higher the cooperativity of the macromolecule. Information on the cooperative nature of the binding curve must be contained in the second moment of the binding capacity. Shifting the B curve along the \nx axis does not affect its shape and hence the variance. If the normalized form of the partition function is used, the first moment vanishes and the variance is given by the second moment only. The value of a2 can therefore be thought of as the variance of the normalized form of the binding capacity, shifted along the lnx axis to make its first moment equal to zero. We now calculate a2 explicitly for a cooperative system. First, let us compute the first two moments for the reference system, for which the partition function is given by eq (2.9) and clearly ocx — a2 = . . . = <xN = K. The moment generating function is G((o) = NK~M
na)
(2.94)
sinTrco The first two moments and the variance are
= -In A:
(2.95a)
(In 2 *) = 7L + \n2K 3
(2.95b)
ol = ^
(2.95c)
The first two moments are independent of N and depend solely on the binding constant K. The variance is independent of N and K, as expected, and reflects an important property of the B distribution in the reference system. The value of ol = TT2/3 sets the boundary between positive (a 2 < ol) and negative (o2 > ol) cooperativity. In the general case
2.4 Properties of the binding capacity
85
where the macromolecule has a partition function such as eq (2.37), or equivalently eq (2.78), the generating function is given by (2.88) and we have TV
Upon expressing each coefficient a in polar coordinates in the complex plane as a = r expi
°2 = V + 7 ^ 2 2 3
2N2i=ij=i
ln2
- " ±2ti = °l+ o\ - o\ Tj
(2.97)
Nj=i
which embodies the mathematical 'driving forces' for cooperativity. There are three components that determine the value of a2. The first component is the variance of the reference system. The second and third components are derived from the distribution of r and cp values associated with the TV coefficients a in the partition function (2.78). The second component, o>> is the variance of the r values, while the third component, o\, is the variance of the cp values. These variances oppose each other and their net balance determines whether the variance of the BJN distribution in the general case is greater or smaller than that of the reference system. Cooperativity is determined by the balance of these two opposite 'forces'. The variance of the norm of the coefficients, o2r, tends to increase the value of o2 with respect to o\. The variance of the angles formed by the a in the complex plane, o\, tends to decrease the value of a2 with respect to CJQ. Therefore a necessary, although not sufficient, condition for positive cooperativity is that at least two coefficients a in the partition function are complex conjugate. No positive cooperativity is possible if W can be written in factorial form with coefficients that are all real and positive. The necessary and sufficient condition for positive cooperativity is that o\ > o\, i.e., that the dispersion of the angles formed by the coefficients a in the complex plane exceeds the dispersion of their radii. This is always guaranteed if the as are distributed along the unit circle, since in this case o2r is at its minimum value of zero. The maximum value allowable for o\ is exactly o\, or TT2/3, since a2 cannot be negative. On the other hand, negative cooperativity has as a necessary condition that o2r > 0, or that at least two coefficients a have a different norm. The necessary and sufficient condition is that o\ < o2r, i.e., that the dispersion of the angles formed by the coefficients a in the complex plane does not exceed the dispersion
2 Global binding processes
86
of their radii. This is always guaranteed if the as are all real and distinct, since in this case o\ is at its minimum value of zero. Unlike the case of o\, the value of o2r can grow without bounds since a2 remains positive. The distribution of coefficients a in the complex plane shown in Figure 2.19 illustrates practically the connection between the properties of the B distribution and cooperativity. The predictions derived from eq (2.97) are confirmed by the analysis of two special cases of interest. The highest degree of positive cooperativity is reached when all terms of the partition function are negligible compared to those reflecting the contribution of the unligated and fully ligated forms. In this case the partition function tends to the asymptotic form W(x) = 1 + ANxN (2.98) d
a s
-0
B-
-G-
OJ
d
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Re Figure 2.19 Distribution of the coefficients a for N = 4 in the three cases shown in Figure 2.18. Positive cooperativity is linked to complex conjugate values (filled circles), while negative cooperativity is characterized by distinct values on the positive real axis (open circles). In the case shown, the coefficients are equal in pairs. The reference system is depicted as four identical values on the positive real axis (open squares).
2.4 Properties of the binding capacity
87
Although this form is physically unrealistic, it is instructive to analyze the behavior of o2 in such a limit. The coefficients a are given by (* = 0 , l , . . . t f - l ) (2.99) N and map on the unit circle in the complex plane upon a simple conformal transformation. In tthis case o2r = 0 and we need only to consider the distribution of angles cpk+1 =
1
2k
)
(k = 0,l,...N-l)
(2.100)
N Application of eqs (2.96) and (2.97) yields (In*) = -
lL
3N2 J^ 2 3N
^ TV
(2.101a)
lnM
N2 A2 N
(2.101b) (2.101c)
The variance is identical to that of a reference system divided by N2. This is the minimum possible value of a2 for a system containing TV sites. The variance tends to zero in the limit N -» o°, indicating that the binding capacity of a fully cooperative system of infinite size is a (5-function peaked at lnx m = ()nx). In this limit, the as become uniformly distributed on the unit circle with a variance o\ = o\. Hence, the term o\ = TT2/3 in eq (2.97) is the variance of the distribution of the angles when the as are uniformly distributed on the unit circle. It also follows from the foregoing argument that, if B is finite anywhere along the lnx axis, then the as cannot be uniformly distributed on the unit circle. This result states in a form much simpler than the Lee-Yang theorem of statistical mechanics (Lee and Yang, 1952) the impossibility of obtaining a phase transition-like divergence of B (a2 = 0) for finite N, since in this case B is always finite and so is its variance. The other limiting case is observed when the partition function factors out into TV binomial terms with all real and positive coefficients, i.e., W(x) = (1 + axx){\ + a2x) . . . (1 + aNx) (2.102) This situation arises when dealing with strong negative cooperativity, or when all sites are different and independent. In this case o\ = 0 and we need only to consider the distribution of the norms. The first two moments
88
2 Global binding processes
and the variance a2 are in this case (2
N =
£ 3
+
ln2a1 + ln2a2 + N
...+ln2aN
a2 = - + J - V y In2 ^ 3 2N2^F
(2.103c)
The variance grows without bounds with the heterogeneity of the binding affinity of the individual sites. 2.5 Linkage effects: The reference cycle
We now turn to the properties of a macromolecule reacting with two ligands, X and Y. This provides the simplest case of interest in the analysis of linkage effects that form the basis of our subsequent discussion of site-specific binding processes. At constant T and P, a macromolecule containing two distinct binding sites for ligands X and Y is a generalized ensemble whose properties can be derived from the definition of the partition function W(x, y) = Yoo + Y10x + Y01y + Ynxy (2.104) The first index refers to ligand X, whose activity is x and the second index to ligand Y, whose activity is y. Each ligation intermediate is characterized by a mechano-thermal partition function Y,y, related to the Gibbs free energy G/; = — kBTlnYtj. The standard free energy change for the reaction (2.105) M + iX + ;T €> MXtYj is given by AGtj = Gtj -Goo = kBTln^
= -kBTln
A(j
(2.106)
The mechano-thermal partition function Yoo has been set equal to one without loss of generality and the unligated form of the macromolecule is taken as reference. The term Atj gives the overall equilibrium binding constant associated with (2.105). When dealing with linkage effects, we are specifically interested in the way the two ligands affect the binding of each other. Analysis of linkage effects in terms of thermodynamic principles allows us to derive the properties of either ligand from the binding of
2.5 Linkage effects: The reference cycle
89
the other, with the obvious practical advantage that information on either ligand can be derived from a set of a few carefully designed experiments. In the presence of ligand Y, the response functions for ligand X can be derived from the partition function by differentiation, keeping the activity of Y constant. We focus our discussion on ligand X, but of course it makes no difference which ligand is chosen. The function X is AwX + A Xy
X = (™*L) =
«
=
XKX
(2.107)
\ d\nx Jy 1 + A10x + A01y + Anxy 1 + xKx which is the familiar form involving the affinity function. The function XK is given by (2.108) 1 + A01y It is independent of x, as expected for a macromolecule containing a single site for ligand X, but is a function of y. The effect of the second ligand can be quantified by studying how the affinity function of ligand X changes with the ligand activity of Y. In the absence of Y, the affinity function of X is equal to A10. In the presence of saturating concentrations of Y, the affinity function of X becomes equal to An/A01. Hence, the affinity of X increases with y if An > A01A10, and decreases otherwise. The presence of Y has no effect on the binding of X if An = A01A10. Linkage is reciprocal, as can be seen from consideration of the binding properties of Y in the presence of X. The relevant expression for Y is Y
=
l ] \d\ny
=
= Jx
1 + Awx + A01y + Anxy
1+
y
and the affinity function K is given by An + Anx 1 + A10x X Hence, if increasing y increases K, then an increase of x must increase y K and vice versa. The condition for linkage between X and Y is one and the same for both ligands and depends solely on the value of the product A0iA 10 compared to An. The significance of these terms can be understood from the definition (2.106). We now introduce the free energy of coupling (Weber, 1975, 1992; Hill, 1977) (2.111) AGC = AGu - AGQI - AG10 If the free energy change for binding X and Y is different from the sum of the free energies of binding of X and Y separately, then binding of either yK
=
90
2 Global binding processes
ligand must affect the binding of the other. Linkage is observed whenever A G c ^ 0 . Specifically, A Gc < 0 positive linkage (2.112a) AGC > 0 negative linkage (2.112b) AGC = 0 no linkage (2.112c) The significance of the foregoing arguments becomes clear upon examination of the reference cycle depicted in Figure 2.20. The binding reactions of X and Y to the macromolecule are depicted as edges of a cycle with the ligation intermediates located at the vertices. Let °KX and lKx be the binding affinities of X to the unligated and Y-ligated forms of the macromolecule and, likewise, let °^Y and XKY be the analogous quantities for Y. Then, by definition
B£2
(2.113a)
m*l
(2.113b)
[M]x
[MY]*
^
'
(
[M]y 1 i/-
[IVLA. I J
/^ Y
• ~ 11 o j \
(Z.11.3(1)
[MX]y
Hence, A x AY -
AY A x
(/.114J
Of the four reactions depicted in Figure 2.20, only three are independent and the fourth one can be derived unequivocally from the others. The connection between the parameters defined in the reference cycle and the coefficients of the partition function is derived in a straightforward way and is given by °#x = A10
M
MY
°Kxx <=>
<=> l Kx x
(2.115a)
MX
MXY
Figure 2.20 The reference cycle.
2.5 Linkage effects: The reference cycle
91 (2.115b)
°KY = Aoi
(2.115c)
IKY
(2.115d)
=- ^
A io
The free energy of coupling is then l
l
K
AGC = -kBTln—-
K
= -k BT\n—-
(2.116)
and is given by the difference in the free energy of binding of either ligand induced by the presence of the other. If the binding affinity increases in the presence of the second ligand, then AGC < 0 and so forth for the other cases. The affinity function plays an important role in the analysis of linkage effects in the particular case of the reference cycle. Consider the derivative of lnxK in eq (2.108) with respect to the ligand activity y of Y din* A: = dlny
A ny
_
A01y 1 + AOi.y
A io + Any
This is a key thermodynamic property whose significance can be understood as follows. The four configurations in the reference cycle can be split into two manifolds, one reflecting the binding of Y to M to yield MY, and the other reflecting the binding of Y to MX to yield MXY. Each manifold can be assigned an appropriate partition function = 1 + °KYy
(2.118a)
= 1 + l KY y
(2.118b)
The configuration M is used as reference in the manifold {M, MY}, while MX is used as reference in the manifold {MX, MXY}. The connection with the partition function of the system as a whole is provided by the relationship W(x, y) = °V(30 +lxP(y)°Kxx
(2.119)
which is identical to (2.104), by virtue of eqs (2.115a)-(2.115d) and the relationship between the As and the Ys. The index 0 or 1 refers to the ligation state of the macromolecule with respect to X. The function Y in the two manifolds is y
=
dln^ dlny
=
KYy l
1 + KY y
=
Any A10 + Any
92
2 Global binding processes
dln*P = »K Yy = A01y dl dlny l + °KYy 11++ AA01y These expressions are identical to those at the right-hand side of eq (2.117) so that oy =
dlny The change in the function lnxK gives the difference in Y-ligated sites between the two manifolds. Since the two manifolds represent the unligated and X-ligated configurations of the macromolecule, eq (2.121) expresses the change in ligation with respect to ligand Y when ligand X is bound, or the net number, AY, of molecules of Y exchanged upon binding of X. If AY > 0, binding of Y facilitates binding of X and the affinity for X increases with increasing y. On the other hand, if AY < 0, binding of Y opposes binding of X and the affinity for X decreases with increasing y. Finally, if Ay = 0, binding of Y has no net effect on binding of X. The trivial case where Y does not bind to the macromolecule is included in this third possibility as a special case. The function lnxK for the interaction of thrombin with the inhibitor hirudin is plotted in Figure 2.21 as a function of the logarithm of Na + concentration. Hirudin represents ligand X in the reference cycle, while Na+ plays the role of ligand Y (Ayala and Di Cera, 1994). Increasing the concentration of Na + , at constant ionic strength, increases the binding affinity of hirudin, so that the two ligands are positively linked. The slope of ]n*K is given in Figure 2.22 and represents the net number of Na + ions taken up upon hirudin binding. Analysis of the data in Figure 2.21 according to eq (2.108) yields all the coefficients of the partition function (2.104). The free energy of coupling for the thermodynamic cycle involving hirudin and Na+ binding to thrombin is AGC = —1.7 ± 0.1 kcal/mol, which indicates that hirudin and Na+ enhance each other's binding free energy by 1.7 kcal/mol. This is a paradigmatic example of the power of approaching macromolecular interactions in terms of linkage effects. The binding properties of Na+ which are difficult to measure directly are derived through the effect on the binding properties of hirudin. Hence, binding of a ligand X to the macromolecule contains information not only on the properties of X, but also on any other process linked to the binding of X. The self-consistency of the results obtained from linkage studies can be checked by direct measurements of binding, when available. In the case of Na+ binding to thrombin, such measurements are shown in Figure 2.3 as determined by fluorescence titration (Ayala and Di Cera, 1994). A
2.5 Linkage effects: The reference cycle
93
q CO
W) O
IT) r^
IT)
O
-4.0
-3.0
-2.0
-1.0
0.0
log [Na + ] (M) Figure 2.21 Linkage between hirudin and Na+ binding to thrombin (Ayala and Di Cera, 1994) represented as the change in the logarithm of hirudin binding affinity as a function of the logarithm of Na + concentration. Experimental conditions are: 5 mM Tris, 0.1% PEG, I = 0.2 M, pH 8.0 at 25 °C. The curve was drawn according to eq (2.108), with parameter values A^ = 1.1 x 1011 M"1, A01 = 34 M"1, An = 7.2 x 1013 M"2. value of K = 35 M" 1 is derived for the binding constant of Na + to the free enzyme. This parameter is the same as A01 in the reference cycle and agrees extremely well with the value derived independently through linkage studies of the effect of Na + on hirudin binding. A special case of the reference cycle that is of particular interest arises when binding of either X or Y to the macromolecule excludes binding of the other ligand. This situation arises when the two ligands share the same site or portions of it, or when they bind to distinct sites that affect each other through substantial conformational transitions of the system. The difference between these cases deserves much consideration in practical applications. The ligated intermediate MXY is automatically excluded
2 Global binding processes
94 00
d
CO
d
%>
* 2
I q d -4.0
-3.0
-2.0
-1.0
0.0
+
log[Na ] (M) +
Figure 2.22 Net number of Na ions exchanged upon hirudin binding to thrombin. The curve is the derivative of the data shown in Figure 2.21 and was drawn according to eq (2.121). from both the cycle and the partition function, since An = 0. The two ligands are negatively linked in this case. Hence, X
K =
A01y
(2.122)
The inverse of XK increases linearly with the concentration of Y. This situation is illustrated in Figure 2.23 for the competition of hirudin and Cl" binding to thrombin. The slope in the affinity plot (2.123) = -°Y = AY dlny gives the number of Y molecules exchanged upon binding of X and is a direct measure of the binding isotherm of Y. In this case, the binding curve of a second ligand (Cl") can be constructed directly using the
2.5 Linkage effects: The reference cycle
95
linkage with the first ligand (hirudin). When direct competition between X and Y cannot be proved experimentally by independent sources, assessment of binding domains for the two ligands may be misleading. In principle, the effect reported in Figure 2.23 can be due to non-competitive inhibition induced by binding of Y to a site completely different from the X binding site. This fact has nontrivial consequences in molecular recognition studies and drug design. The neurokinin-1 receptor provides a convincing and important example. This receptor has competitive-like antagonists that bear no chemical or structural resemblance with agonists, thereby indicating that inhibition is mediated by binding to a different site (Fong, Huang and Strader, 1992; Jensen, Gerard, Schwartz and Gether, 1994; Pradier etal., 1994). q
CO
o CN
LO
d
q d 0.000
0.050
0.100
0.150
0.200
[Cl] (M) Figure 2.23 Linkage between hirudin and Cl binding to thrombin (Ayala and Di Cera, 1994), represented as the change in the inverse of hirudin binding affinity as a function of Cl" concentration. Experimental conditions are: 5 mM Tris, 0.1% PEG, I = 0.2 M, pH 8.0 at 25 °C. The curve was drawn according to eq (2.122), with parameter values Ai0 = 2.2 x 1012 M"1, AOi = 12 M"1.
96
2 Global binding processes
We will now use the reference cycle to illustrate the importance of arguments developed in Section 1.3. At constant T and P, the thermodynamic potential of interest is IT and specifically (2.124) d n = - I d f t - Y d/iY = - A ^ r d l n ^ F ^ x , JUY) The reciprocal nature of the linkage between X and Y is a consequence of the first law of thermodynamics and the nature of exact differential of dll. The Euler relationship (2.125) indicates that if ligand Y facilitates binding of ligand X (3X/3//Y > 0), then ligand X must facilitate binding of ligand Y ( 9 Y / 9 J U X > 0 ) . The relationship above can be rewritten in the more useful form for practical applications (Wyman, 1948, 1964)
(j*-\
. (JL.)
(2.126)
\d\nyjx \dlnxjy since T and P are constant. The matrix G' associated with the potential —IT is equal to
G' = I8}
8
}\
(2.127)
where the indices 1 and 2 refer to ligands X and Y respectively. The matrix G' is two-dimensional instead of four-dimensional, since T and P are held constant and make no contribution to dn. The diagonal elements of G' give the binding capacities of X and Y „ /92 lnW\ (1 + Aoly)(Alo + Any)x Bx = = = X(l - X) 2 2 \ > ' (1 + A10x + A01y + Anxy) (2.128) _ . ^ in x . _ D uY - i -i
\dlny2/x
v- • A10x)(A01
+ Anx)y
(1 + A10x + A01y + Anxyf
_ - r ( l - Y)
(2.129) These quantities are always positive, as implied by the second law of thermodynamics. The off-diagonal elements give the linkage coefficients, which are identical by virtue of the linkage relationship (2.125) and are explicitly given by 2 (An - AwA0l)xy (2 m ) = ( 8 lnW \ \31nv31nx/* dlnydlnx Jx (1 + A10x + A01y + Anxy)
2.5 Linkage effects: The reference cycle
97
(An - AwA0l)xy I d\nxd\ny ) y (1 + A10x + A01y + A The linkage coefficient can be positive, negative or zero depending upon the value of the difference in the numerator. This difference defines the cutoff for positive, negative or no linkage between the ligands, as already seen. Also, the linkage is always negative if An = 0, i.e., if binding of either ligand excludes binding of the other. The determinant associated with the matrix G' is given by the expression detG' = xy Any)(l
X
+ Anx) - (An - A10A01)2xy
+ A10x)(A01
(1 + A10x + A01y +
Anxyf (2.132)
The second law of thermodynamics demands that detG' is always positive. Inspection of the numerator proves that this is indeed the case. The fundamental consequences of thermodynamic stability dealt with in Section 1.3 can now be understood in terms of the elements of the matrix G'. The binding capacity of ligand X, as defined in eq (2.128), is computed at constant chemical potential of ligand Y. The same quantity can be computed at constant Y and the results compared. The second law states that Bx computed at constant y is never smaller than Bx computed at constant Y. The same holds for .By computed at constant x o r l Consider the definition of W for the reference cycle analogous to eq (2.119) W(x, y) = °W(x) + ^(xfKvy
(2.133)
with °W(x) = 1 + °Kxx
(2.134a)
M x ) = 1 + l Kx x
(2.134b)
and l x
=
dln^W dlnx
=
H In °VL/
>Kxx 1 + xKxx
Anx
=
Then
^
AQX
°KvX
ox - ^i2-Zl = — ± 2 ^ _ = dlnx
(2
1 + °KX X
X = °X(1 -Y)
" ^
1 + ^lio-t
+1XY
(2.135b) (2.136)
98
2 Global binding processes
The binding capacity Bx at constant Y is derived from differentiation as B'x = °B(1 - Y) +XBY (2.137) where °B andlB are the binding capacities of X when the site binding Y is free or bound respectively. On the other hand, the binding capacity Bx at constant y is Bx = °J3(1 - Y) +XBY + LXY{XX - °X) = B'x + L X Y C^ - °^0 (2.138) Hence, Bx differs from Bx by an additive term involving the linkage coefficient LXY multiplied by the difference lX — °X. The sign of this difference is the same as the sign of An - A10A01. However, this expression also determines the sign of LXY, as implied by eq (2.130). Hence, LXYO*
- °*) ^ 0
(2.139)
This result is readily understood from a physical point of view. If LXY > 0, binding of X is facilitated by binding of Y and consequently XX > °X. On the other hand, if LXY < 0, binding of X is opposed by binding of Y and consequently lX < °X. In either case, the nature of the linkage between X and Y leaves the sign of expression (2.139) unchanged. Hence, Bx & B'x
(2.140)
as implied by the second law of thermodynamics. The special condition Bx = Bx applies in the absence of linkage. Similar arguments hold for the binding capacity of ligand Y. The foregoing analysis leads to the conclusion that the cooperativity of the reference system can be controlled in terms of thermodynamic stability according to eq (2.140) in a model-independent fashion. If the binding properties of ligand X are studied by keeping Y constant, then the binding capacity of X is lower than that observed at constant y. A decreased binding capacity implies the presence of apparent negative cooperativity in the reference system, which is by definition noncooperative if y is kept constant. This result is embodied by the properties of the matrix G' and is a consequence of Hadamard's inequality involving the metric coefficients gf (Gantmacher, 1959). The binding capacity Bx is the diagonal element gn of G' in (2.127), and Bx is the value of g'n 'contracted' over the manifold spanned by ligand Y, i.e., '
g
822 2
detG'
(
(
g22
Hence, Bx = g{t > g'n = Bx, due to the Euler relationship g[2 — g2\-
2.6 Linkage effects: Two Hgands
99
2.6 Linkage effects: General treatment involving two ligands
Some features of the foregoing treatment of the reference cycle bear on the more general case of an arbitrary number of sites for ligands X and Y. The two-dimensional array of possible ligated species in the general case is sketched in Figure 2.24. This array can be seen as an ensemble of reference cycles each composed of a 2 x 2 array of ligated species. It is, however, more appropriate to consider the ensemble of species in terms of separate manifolds for X and Y binding to particular Y- or X-ligated configurations. The partition function for this generalized ensemble at constant T and P is analogous to eq (2.104), except that the double summation extends to N sites for X and M sites for Y, i.e.,
nx,y)
= J,^Yijxy
(2.142)
/=0;=0
Again the first index refers to ligand X, whose activity is x9 and the second index to ligand Y, whose activity is y. Each ligation intermediate is characterized by the mechano-thermal partition function Y,y and the Gibbs free energy Gi} = — kBTinYy. The standard free energy change for the reaction M + iX + /Y O MX/Yy
M
<=> MX
U MY
U <=> M X Y
I MY2
<=> M X Y 2
<=> M X 2 Y
Figure 2.24
<=>
<=> M X 2 Y 2
<=>
<=> M X ^ Y
fl <=>
<=> M X ^ Y 2
H
I
«
^
%
t
t
<=> MXYM
»
B
t
I
MYM
<=> MX 2
(2.143)
MX 2YM <=>
<=>
Linkage scheme for the binding of two ligands.
100
2 Global binding processes
is given by
AGtj = Gtj - Goo = V l n —
oo
=
-kBTlnAtj
(2.144)
These equations are identical to eqs (2.105) aid (2.106), except for the fact that they apply to the general case of i = 0, 1, . . . N and / = 0, 1, . . . M. As usual, all equilibria can be recast in terms of the overall constants Aij letting YOo = 1 without loss of generality. This is equivalent to selecting M in Figure 2.24 as the reference configuration. The partition function (2.142) is therefore iV M
vi*, y) = ^Ajxy
(2.145)
i=0;=0
The functions X and Y can be derived from W by differentiation as follows N M
(2.146)
\ 31nx )y i=Oj=O N M i=Oj=O
(2.147)
d\ny i=0/=0
The binding capacities of the two ligands are N M
„
N M ,*W\ 2
--**-)
\dlnxh
=-
i=0/=0 N M
/=0/=0 N M
N M
\ i=Oj=O N M
i=oy=o
dlny
M i=Oj=O
(2.148)
M
(2.149)
\ i=0j=0
We know from Section 2.2 that X can be written in terms of the affinity function as in eq (2.42), regardless of the form of the partition function.
2.6 Linkage effects: Two ligands
101
The affinity function for the general case of the linkage between two ligands is given by N M
(2.150)
N M
;=o/=o N M
yK = _J=*1
(2.151)
N M
i=0j=0
In order to clarify the significance of eqs (2.150) and (2.151), it is useful to consider the ensemble of species in Figure 2.24 in terms of separate manifolds, as done in the case of the reference cycle. Each column of the two-dimensional matrix contains the ligated intermediates for Y binding to a particular X-ligated configuration. In general, the rth column contains all possible Y-ligated configurations of the rth X-ligated species. Each column is a manifold which can be characterized in terms of a generalized partition function. Therthcolumn has a partition function M
A
7=0
The form of the coefficients of eq (2.152) is readily understood in view of the fact that the rth X-ligated species, MX/9 is used as reference in the rth manifold and this species contributes a term AiOxl to W. If all terms of the rth manifold are expressed relative to MX,-, a polynomial expansion in the variable y alone is obtained as in eq (2.152). The partition function of the system as a whole can be rewritten as follows W(x,y) = JjiW(y)AiOxi
(2.153)
Using the manifolds relative to each X-ligated intermediate, we have essentially grouped configurations with the same number of X-ligated sites in Figure 2.24 into ad hoc partition functions, and expressed ^ as a polynomial expansion in x with coefficients written in terms of these partition functions. It should be pointed out that, although Aoo = 1 since the species M is used as reference in l F, the partition function °^/(y) is not equal to one. In fact, this term contains all intermediates of the first
102
2 Global binding processes
column in Figure 2.24. If this partition function is factored out, the resulting polynomial expansion W(x, y) = °W(y)[l + yAxx + yA2x2 + . . . + yANxN]
(2.154)
differs from eq (2.37) in the prefactor, which makes no contribution to the derivation of the properties of X since it does not depend on x. Also, the M s are functions of y as follows yAi
=
i
The coefficient yAt represents the overall equilibrium constant for the reaction M + iXo MX, in the presence of ligand Y. Had we defined manifolds of X-ligated species for given Y-ligated intermediates, we would have obtained a set of coefficients xAj for the reaction M + y'Y o MY ; in the presence of ligand X. In either case, the definition of partition functions over each manifold of interest 'contracts' the original two-dimensional array of species in the linkage scheme in Figure 2.24 into a one-dimensional array of species with coefficients depending on the activity of the second ligand. Measurements of the coefficients yA as a function of y reveal important properties of the system. The derivative = lY -°Y
= AYt (2.156) dlny shows that the change in lnMj gives the difference in Y-ligated sites between the ith and zeroth X-ligated species. The quantity AY( is the net number of Y molecules exchanged when / molecules of X bind to the X-unligated form of the macromolecule. The sign of AY( denotes the nature of the linkage involved in binding / molecules of X. If AYt>0y binding of i molecules of X to the X-unligated form of the macromolecule leads to an increase in the number of Y-ligated sites. In this case the two ligands are positively linked. On the other hand, if A Yt < 0, binding of i molecules of X to the X-unligated form gives rise to a decrease in the number of Y-ligated sites and binding of either ligand opposes binding of the other. The case AYt = 0 denotes that binding of / molecules of X has no net effect on the amount of ligand Y bound to the macromolecule. Once all values of AYt are known, it is possible to dissect the linkage between X and Y at the level of stepwise reaction steps. In fact, the net number of Y molecules exchanged upon adding one molecule of X to the (/ — l)th X-ligated intermediate is clearly (2.157)
2.6 Linkage effects: Two ligands
103
If all values of Ayt are added up from i = lto i = N, the total net number of Y molecules exchanged in fully ligating the macromolecule with X is derived as Ayx + Ay2 + . . . + AyN = AYN
(2.158)
y
The overall This quantity is the same as the derivative dln AN/d\ny. equilibrium constant yAN is related to the mean ligand activity for X as follows yXm = yA-N1/N
(2.159)
Hence, ***=. _ -**!. _ - A , (2.160) dlny N The change in the mean ligand activity of X, which can be derived from X or Bx by numerical integration, yields an important property of the system, i.e., the net number of Y molecules exchanged on the average per site ligated to X. The linkage between two ligands may affect the cooperative properties of the macromolecule. Assume that each column in Figure 2.24 depicts a reference system. In other words, let the binding of Y to the rth X-ligated intermediate be described by the simple polynomial expansion l
W(y) = (1 + Kiy)M
(2.161)
We assume that A^AJQ = CjMK\ in eq (2.152). Then, let the first row in Figure 2.24 be itself a reference system, so that binding of X to the macromolecule in the absence of Y occurs with no cooperativity, or Ai0 = CiNK\ It follows from eq (2.155) that yA^c^m
/ 1 + Kv \M *-)
(2.162)
Although binding of Y to each X-ligated intermediate is noncooperative, and binding of X to the Y-unligated form is noncooperative, binding of X in the presence of Y becomes cooperative if Y binds with different affinity to the different X-ligated intermediates. Cooperativity in the binding of X arises as a result of the linkage between X and Y, only requiring a different binding affinity for different manifolds. Linkage between X and Y translates into linkage between X and X. If binding of X progressively increases the affinity of Y, say Kt > Kt-l9 then, after the ith X molecule binds, binding of Y is favored. This, in turn, favors the binding of other molecules of X. The resulting positive cooperativity in X binding is a consequence of the effect of X on Y, and the reciprocal effect of Y on X,
104
2 Global binding processes
which results in an effect of X on the binding of itself. This is the thermodynamic basis of allosteric transitions (Wyman and Allen, 1951; Wyman, 1967), whereby cooperativity is generated indirectly through linkage effects using either a second ligand, or multiple conformational states (Monod, Changeux and Jacob, 1963; Monod, Wyman and Changeux, 1965; Koshland, Nemethy and Filmer, 1966). Aspects related to thermodynamic stability can be obtained in a straightforward way as was done for the reference cycle. The properties of the matrix G' depend only on the number of ligands involved and not on the number of sites. Hence, the inequality Bx > Bx in the presence of linkage between X and Y is a direct consequence of the Hadamard inequality involving the metric coefficients g' and holds quite generally for an arbitrary number of sites for binding of X and Y. Other important aspects arise when the two ligands compete for the same site, or share portions of it. In this case of 'identical linkage' (Di Cera, Doyle, Connelly and Gill, 1987a; Wyman and Gill, 1990), binding of either ligand excludes binding of the other. In the reference cycle, identical linkage is characterized by a negative value of the linkage coefficient LXY. Intuitively, one would expect LXY < 0 also in the general case, due to the competitive nature of the linkage. However, this is not the case and under certain conditions LXY can change sign with saturation. The partition function for two identically linked ligands is given by (2.163) i=0/=0
and is identical to eq (2.145), except for the fact that all ligated intermediates MX,-Yy with i + j > N are eliminated from the scheme and the double summation. The linkage coefficient LXY is derived by differentiation as follows TV N-i
_ 32lnxif(xy
y) _ i=oj=o
N N-i
N N-i
_i=oj=o
i=oj=o
N N-i i=0j=0
\i=0j=0
(2.164) The sign of LXY is determined by the sign of the expression N N-i
N N-i
N N-i
N N-i
i=0j=0
i=Oj=o
i=0j=0
i=Oj=O
(2.165)
2.6 Linkage effects: Two ligands
105
For N = 1, the first term vanishes and the linkage is always negative, as already seen for the reference cycle. For TV = 2, one has L(x, y) = xy{An
- A10A01
- 2A10A02y - AnA02y2
- AnA20x
2
-
2A01A20x
- 4A20A02xy - A20Anx2)
(2.166)
The sign of L{x,y) is negative, except for small values of x and y, in which case it depends on the sign of the difference An — A 10 A 0 i. If An > A10A0U L(x,y) is positive and so is the linkage coefficient LXY> notwithstanding the fact that binding of X opposes binding of Y and vice versa. The case An> A10A01 corresponds to the situation where the two ligands facilitate binding of each other when they are both bound to the macromolecule. At low concentrations of X and Y the dominant species are M, MX, MY and MXY, and MXY is favored due to the condition An > A10A01. At low concentrations, both ligands can bind to the macromolecule due to the presence of multiple sites, and they can do so with positive linkage. On the other hand, increasing the concentration of one ligand leads to an increase of the amount bound and a necessary displacement of the other ligand. Hence, at high saturation the linkage between X and Y is always negative, regardless of the value of An. This result is true in general for arbitrary N. The case of oxygen and carbon monoxide binding to hemoglobin is a beautiful example of how ligands competing for the heme sites can favor each other's binding at other sites. At low oxygen saturation, an increase in carbon monoxide partial pressure first causes a 'paradoxical' increase of the amount of oxygen bound at other sites, and then leads to total oxygen desorption (Douglas, Haldane and Haldane, 1912; Di Cera etai, 1989). We conclude our analysis of linkage effects with a case of particular importance in practical applications. In the general treatment involving two ligands we have assumed that all response functions for one ligand can be derived by keeping the activity of the other ligand constant. However, in many cases a change in the chemical potential of ligand X cannot be obtained experimentally without a concomitant change in the activity of Y. This is especially true when water represents the second ligand and hydration effects are involved in structural changes of functional significance (Timasheff, 1990; Colombo, Rau and Parsegian, 1992). Consider the case of ligand X and water W (ligand Y) binding to the macromolecule under conditions where both X and W are in large excess with respect to M. This is the situation most often encountered in practice. The Pfaffian form of the potential for the generalized ensemble at constant T and P is d n = -kBT(X
din x + Wdlnw)
(2.167)
106
2 Global binding processes
The result above implies that the macromolecule is taken as our system, as we know from arguments discussed in Chapter 1. Since X and W are in large excess compared to M, the chemical potential of the macromolecule, dll, is expected to make only a small contribution to the properties of the solution. The Gibbs-Duhem equation for a solution containing X and W alone is 0 = -kBT(nxdlnx + nwd\nw) (2.168) where nx and n w are the mols of X and water present. Due to the fact that the chemical potential of the macromolecule is negligibly small compared to the terms in eq (2.168), a change of x in eq (2.167) will inevitably induce a change of w9 because of eq (2.168). Hence, the number of ligands X bound to the macromolecule must be calculated as the total derivative dlnll/dlnx, rather than the partial derivative (3 In n/3 In ;*;)„,, since w cannot be kept constant while changing x. Substitution of din w in eq (2.167) from eq (2.168) yields the derivative dlnn = x _ m^w (2.169) dlnw This important result obtained by Tanford (1969) shows that the response function X\ measured as the apparent number of ligands X bound to the macromolecule, equals the number expected if the water activity were constant, X, minus a term that includes the number of water molecules bound and the mole ratio of ligand X relative to water, mx = nx/nw. In the case of ligands binding in the /iM-nM range, as usually found in practice, the correcting term in eq (2.169) is negligible and the effect of changing the water activity can be neglected. However, in those cases where ligand X is used in the mM-M range, as in the case of salts (von Hippel and Schleich, 1969; Record, Anderson and Lohman, 1978; Anderson and Record, 1993; Record and Anderson, 1995), or denaturants (Schellman, 1978, 1987, 1990b, 1994; Tanford, 1970), the term mxW may have a significant influence on the measured value of X' relative to X. The value of X' is the preferential binding (Tanford, 1969) defined as the amount of ligand X bound to the macromolecule in excess of the amount 'bound' to the solution. Ligand X binds to the macromolecule in the usual sense only if X largely exceeds mxW in eq (2.169). In the case of X = mxW', the number of ligands bound to the macromolecule per water molecule bound is the same as the mole ratio of ligand X relative to water in solution and there is no 'preference' for the macromolecule relative to the solution. Preferential binding can also assume negative values if the ligand is 'excluded' from the macromolecule (X < mxW) relative to the x
,
=
2.6 Linkage effects: Two ligands
107
solvent. The concept of preferential binding emphasizes the importance of hydration in macromolecular binding and the role of water as the ubiquitous 'second ligand'. An important generalization of the treatment of preferential interaction phenomena has been derived by Record and Anderson (1995).
3 Local binding processes
We are now in a position to deal with the main subject of this monograph, that is, the description of binding and linkage effects arising locally at the level of individual sites of a multi-site macromolecule. Consider a macromolecule containing N binding sites for a ligand X. We seek to resolve the thermodynamics of site-specific effects arising when binding of X can be monitored at each site separately. If the N sites are independent, then the description of site-specific effects becomes a trivial task, since each site behaves according to the rules outlined for a reference system in Section 2.1. When the sites are not only independent but also alike, then the macromolecule as a whole is a reference system. The case we are interested in is the one where all sites are different and linked. In this instance each separate site of the macromolecule is a subsystem open to interactions with other subsystems. Hence, the behavior of an individual site coupled to the rest of the macromolecule is expected to be rather complex, as it must reflect not only the binding properties of the site under consideration, but also the result of interactions with other sites.
3.1 The reference cycle and site-specific cooperativity
The simplest system of interest in the discussion of binding processes at the local level is that composed of a macromolecule M containing two binding sites for ligand X. Again we assume T and P constant and that the macromolecule does not change its aggregation state. In the global description the partition function of such a system is given by W(x) = Yo + Yi* + Y2x2 = 1 + Axx + A2x2 (3.1) The significance of the Ys and As in eq (3.1) has already been dealt with at length. The global description tells us that for a system of two sites 108
3.1 The reference cycle
109
there are two independent coefficients to be determined experimentally. The cooperativity is positive or negative depending on the sign of the expression A = AA2 - A\ = Akx{k2 - kx) (3.2) with A = 0 providing the noncooperative case. The key question to be addressed in what follows is whether the global, macroscopic condition for cooperativity, A =£ 0, reflects the true pattern of interaction between the sites. The partition function (3.1) embodies the overall behavior of the system and as such all quantities derived from it reflect properties of the system as a whole. When the behavior of constituent sites becomes the focus of our investigation, other properties of the system, more specific and local, are taken into account. Analysis of these properties reveals that the global description provides a gross simplification of the communication among the sites. Cooperativity patterns in the global description can underestimate, or even distort, the effective nature of the interactions. The importance of the thermodynamic treatment of site-specific effects stems entirely from the possibility of decoupling linkage and cooperativity in a direct way. Consider the reference cycle in Figure 3.1. A comparison with the cycle in Figure 2.20 reveals an important parallel to be discussed in detail in Section 3.3. The binding reactions of the two ligands in Figure 2.20 have been replaced by the reactions of ligand X to site 1 and 2 of the macromolecule. The substitution is perfectly valid, since the properties of the reference cycle in Figure 2.20 do not depend on the nature of the ligands. Hence, X and Y in Figure 2.20 can be treated as identical to give rise to the cycle in Figure 3.1. We now write down the partition function for the reference cycle in Figure 3.1, using arguments developed in Section 2.5, as follows W(x) = Yoo + (Y10 + Y01)* + Ynx2 = 1 + (A10 + A01)x + Anx2 (3.3)
K2
OO
o
»0
cl2K2 Figure 3.1 The reference cycle for site-specific binding to a macromolecule containing two sites.
110
3 Local binding processes
The indices refer to sites 1 and 2 and the Ys and As were defined in Section 2.5. Specifically, Y10 is the mechano-thermal partition function of the configuration with site 1 ligated and site 2 unligated, A10 is the overall equilibrium constant for X binding to site 1 when site 2 is unligated and so forth. The unligated form of the macromolecule is taken as reference as usual, which makes YQO = 1. We know from Section 2.5 that, out of the four reactions in the cycle in Figure 3.1, only three are truly independent. Consistent with this fact, the partition function (3.3) contains three independent coefficients. However, this partition function must be equivalent to eq (3.1), which describes the properties of the same system. Hence, the question arises as to how many independent parameters are to be considered in the analysis of a two-site macromolecule. The global description requires two, but the approach depicted in Figure 3.1 based on the local, site-specific description demands three. The contradiction is only apparent. In fact, the coefficients A10 and A01 in eq (3.3) are coupled to the same power of x, so that Ax = A io + A01
(3.4a)
A2 = A n
(3.4b)
What the apparent contradiction means is that it is not possible to derive uniquely the site-specific parameters A10, A01 and An from knowledge of the global parameters Ax and A2. On the other hand, if the site-specific parameters are known, then the global parameters can be determined uniquely. Hence, the global description is incapable of providing information on what goes on at the level of individual sites. This fact has been recognized for a long time and the need for a local description of binding processes finds its origin in the early work on the dissociation of polyvalent substances (Wegscheider, 1895; Adams, 1916; Simms, 1926; Edsall and Blanchard, 1933; Edsall and Wyman, 1958). The description of site-specific effects demands resolution of more parameters than the global description can yield. With this important conclusion in mind, we proceed to examine the effect on cooperativity. It is convenient to define new site-specific parameters that will henceforth be used. The equilibrium binding constants A10 and A01 reflecting the binding of X to either site when the other site is unligated will be designated Kx and K2 respectively. The binding constant An is defined as the product c 1 2 ^ i ^ 2 , where c12 expresses an interaction constant between the two sites. It is particularly important to understand the thermodynamic significance of these parameters. The binding constants Kx and K2 are related to the standard free energy changes
3.1 The reference cycle
111
x = -jfc B rin — = -kBT\nKx
(3.5a)
AG2 = -kBTln — = -k BTlnK2
(3.5b)
Y
oo
that measure the difference in the contribution of the mechano-thermal partition functions of each ligated intermediate relative to the reference configuration. It should be stressed that the reference configuration is the unligated form of the macromolecule and therefore Kx measures the binding affinity of site 1 in the macromolecule when site 2 is kept unligated. The interaction constant c12, on the other hand, is a measure of cooperativity between the sites. In fact, it is easy to show from eq (3.3) that c12 =
YooYn
=
An
(3.6) YY AA The free energy change associated with c12 is therefore the free energy of coupling for the cycle in Figure 3.1, as already demonstrated in Section 2.5 with eq (2.111). In other words, AG12 = — kBT In cu = A G n — AG10 — AGOi (3.7) = A G n - AGx - AG2 = AGC The significance of c12 stems exactly from its connection with a thermodynamic coupling parameter and can be considered the equilibrium constant for the 'dismutation' M10 + M10 <=> MQO + M u (3.8) as follows directly from eq (3.6). The partition function rewritten in terms of Ku K2 and c12 assumes the simple form W(x) = 1 + (KX + K2)x + c12KxK2x2 (3.9) and comparison with the global form (3.1) yields Ax = Kx + K 2 (3.10a) A2 = c12K1K2 (3.10b) The condition (3.2) for macroscopic cooperativity written in terms of site-specific parameters becomes A = 4 c 1 2 K 1 # 2 - ( K 1 + K2)2 (3.11) Since A is second-order homogeneous in the Ks, scaling Kx and K2 by the same constant factor leaves the sign of A unchanged. Hence, the sign of A
112
3 Local binding processes
must depend solely on the ratio KXJK2, or equivalently K2/K1. Letting p = K1/K2 yields A = K\[Ac12p - (1 + p)2] (3.12) The sign of A depends solely on the expression in brackets. Symmetry allows us to study eq (3.12) in the range 0 ^ p ^ 1 only. In fact, p > 1 is equivalent to swapping Kx and K2 and defining a new parameter p' = K2/K1 < 1, which leaves the expression in brackets unchanged. The cutoff between positive and negative macroscopic cooperativity is (3.13) 4c12p - (1 + p)2 = 0 This condition for the absence of macroscopic cooperativity does not coincide with the analogous microscopic condition c12 = 1, which implies that the sites bind ligand X independent of one another and the free energy of coupling is zero. The two conditions coincide only if p = 1, or if the two sites are identical and independent as when the macromolecule is a reference system. When p < 1, values of c12 > 1 may yield A = 0 as well. The relationship between microscopic and macroscopic cooperativity is depicted in Figure 3.2. The continuous curve represents eq (3.13) and gives the condition for which macroscopic cooperativity vanishes. The region above the curve is characterized by positive macroscopic cooperativity, while the region below the curve is characterized by negative macroscopic cooperativity. The dashed line c12 = 1 gives the condition for the absence of microscopic cooperativity. The two conditions together define three regions in the c12-p plane. In region A, defined by c12 < 1, there is no ambiguity between macroscopic and microscopic cooperativity. If binding to one site opposes binding to the other site, negative cooperativity is seen at both levels. At the microscopic boundary c12 = 1, the system is always negatively cooperative in the macroscopic sense, unless p = 1. Region B, where (1 + p)2/4p > c12 > 1, is a most interesting one. Here negative cooperativity in the macroscopic description is observed even though the two sites interact in a positive fashion. Region C, where c12 > (1 + p)2/4p, leaves again no ambiguity between macroscopic and microscopic cooperativity. In this region binding to one site favors binding to the other site. The conclusion to be drawn from Figure 3.2 is that macroscopic cooperativity is at most a crude approximation of the true pattern of interaction between the sites. In the case of positive macroscopic cooperativity, interactions are always underestimated if p ^ l . In fact, for any given point in region C such that c12 > 1, there is always a value of p such that A = 0, or even A < 0. This means that, regardless of the strength of
3.1 The reference cycle
113
q
CO
q CN
q d 0.0
0.2
0.4
0.6
0.8
1.0
Figure 3.2 Microscopic and macroscopic cooperativity patterns for N = 2, plotted as the site-specific coupling constant, cu, versus the parameter for binding heterogeneity, p. The continuous line was drawn according to eq (3.13) and represents the boundary between negative (regions A and B) and positive (region C) macroscopic cooperativity. The dashed line provides the boundary between negative (region A) and positive (regions B and C) microscopic cooperativity. In regions A and C microscopic and macroscopic cooperativity coincide at least qualitatively. In region B positive microscopic cooperativity manifests itself macroscopically as negative cooperativity.
positive coupling between the sites, macroscopic cooperativity can be made small, vanishing or even negative by changing the value of p. The heterogeneity of the binding affinities of the sites opposes the effect of the positive coupling between them and tends to cancel, or even reverse, the cooperativity pattern measured macroscopically. As the heterogeneity tends to increase, p—> 0 and A > 0 only if c12 > l/4p. For a system where the affinities of the two sites differ by a factor of 100 (p = 0.01), but binding to one site increases the affinity of the other by a factor of 10 (c12 = 10), one has A < 0 and macroscopic negative cooperativity is
114
3 Local binding processes
measured notwithstanding that the microscopic picture indicates positive coupling between the sites. For this system to show macroscopic positive cooperativity the value of c12 must exceed 25, whilst in a system where p = 0.001 the value of c12 must exceed 250, and so on. The limitations of the global description of cooperative interactions are not confined to the case of positive cooperativity. When negative cooperativity is observed in the macroscopic case, the system can actually be positively or negatively cooperative in the microscopic picture. Particularly interesting is the case c12 = 1, which always leads to macroscopic negative cooperativity if p =£ 1. The heterogeneity of the binding affinities of the sites generates per se a misleading pattern of interaction. In summary, macroscopic cooperativity does not reflect the true pattern of interaction between the sites, unless the sites are identical. If the sites bind with different affinities, then positive cooperativity in the global description always underestimates the coupling between the sites. Negative cooperativity can be totally misleading, since it may result from distortion of positive coupling between the sites, or absence of interactions. In the case of negative coupling between the sites, on the other hand, the interaction may be overestimated in the global picture. Absence of cooperativity in the global description can be misleading as well. Positive coupling between the sites can be canceled by heterogeneity of their binding affinity. The true cooperative nature of the interactions between the sites can only be resolved from analysis of experimental data if the value of c12 is known. This value cannot be obtained from the global description of the facts. The example we have analyzed for the sake of simplicity is paradigmatic for the general case of N > 2, as is intuitively obvious. Without resolving site-specific parameters, our understanding of cooperative phenomena remains limited, if not misleading. As we realize the importance of the local picture, we must abandon the conceptual framework of the global description and enter a different, more complex scenario provided by the thermodynamics of site-specific energetics. The partition function of the system of two interacting sites can be cast in a number of equivalent forms in the local description. The four possible ligation states in Figure 3.1 can be partitioned in two manifolds, as indicated in Figure 3.3. One manifold contains all configurations with site 1 ligated and the other contains all configurations with site 1 unligated. Hence, W(x) = 1 + (Kx + K2)x + cnKlK2x2
= ^ ( x ) + lWx{x)Kxx
(3.14)
Here ° ^ is the partition function of the manifold or subsystem where site
3.1 The reference cycle
£
OO
<=>
115
#O
cnK2
Figure 3.3 The reference cycle for site-specific binding to a macromolecule containing two sites. The four ligation states are partitioned into two manifolds where site 1 is kept unligated (top) or ligated (bottom).
1 is kept unligated and likewise 1}¥ x is the partition function of the subsystem with site 1 ligated. Had we partitioned the linkage scheme in Figure 3.1 according to site 2, we would have obtained V(x) = 1 + (tfx + K2)x + c12K1K2x2 = °W2(x) + l^2{x)K2x Hence, OVU ^v^ 4- 1\I/ (v\ V v
0\I/ (**\ _i_ 1\T/ / v \ f
v
(3.15) /a 1/c\
The importance of defining partition functions for separate subsystems will be made clear in the next section. It is now rather straightforward to arrive at the key thermodynamic quantities reflecting binding to sites 1 and 2 separately. The site-specific binding curve of site 1 is given by the ratio of all configurations with site 1 ligated with respect to the total, i.e., *V (x}
Xx = KlX—^-L
x
®W (x}
= 1- —iii =
K r
21*—
(3.17)
The site-specific binding isotherm Xx can be cast in a form analogous to X in the global description by introducing the site-specific affinity function (3.18) Likewise, for site 2 one has X 2 = K 2x
2
±-L = 1 - -^^-
IT/^v-^
/^2 (1 — X^X
=
VI/^v^ j^2 °\U (Y\
^ — 1 _L x if
(3.19) v
j^
(3.2U) 1 4- K, Y
It is evident from the foregoing definitions that a site-specific binding isotherm is similar to that of a cooperative macromolecule, although it reflects the binding properties of an individual site. This is the
116
3 Local binding processes
consequence of the site being a subsystem open to interaction with the rest of the macromolecule. Many properties of site-specific processes can be derived from analysis of the affinity function using arguments already developed in the global description. For example, the limiting values of XK1 are °K, = Kx (3.21a) -K^enK, (3.21b) Due to coupling between the sites, the affinity of site 1 increases with saturation when c12 > 1 and decreases when c12 < 1. In the absence of true interactions c12 = 1, the affinity function is constant and Xx = _ * £ (3.22) 1 + Kxx The site-specific binding isotherm is identical to the binding curve of a macromolecule composed of a single site, as expected. Similar arguments hold for site 2. The reciprocity of the linkage between the sites is noteworthy. The limiting values of XK2 are °K2 = K2 (3.23a) "K2 = c12K2 (3.23b) and therefore, if binding to site 2 increases the affinity of site 1, then binding to site 1 must increase the affinity of site 2. The derivative of the logarithm of the affinity function is an important linkage quantity. This derivative is given by d In x d In x d In x where XX is the number of ligated sites of the macromolecule, other than site 1, when site 1 is ligated and °XX is the analogous quantity when site 1 is unligated. If lXx = °XX for all values of x, then XKX is a constant and site 1 behaves as an independent site. On the other hand, if lXx =£ °XX binding to site 1 affects the saturation of site 2 and, conversely, binding to site 2 affects the binding to site 1. This translates into a departure from a simple noncooperative binding curve and site 1 behaves 'cooperatively'. The driving force for site-specific cooperativity is provided by the difference 1X1 — °X1 in eq (3.24) and leads to positive cooperativity when l Xi > °X1, or negative cooperativity when 1X1 < °XX. This driving force is completely analogous to that involving two different ligands, as seen in Section 2.5. Hence, the molecule of X binding to site 2 can be seen as a 'second' ligand affecting the properties of the molecule of X binding to site X
3.1 The reference cycle
111
1. The change in the function l n * ^ gives the difference in ligation of site 2 upon binding to site 1, or else the net number of molecules exchanged at site 2 when site 1 is ligated. If XXX — °XX is positive, binding to site 2 facilitates binding to site 1 and the affinity of site 1 increases with ligation. is negative, binding to site 2 opposes On the other hand, if 1X1-°X1 binding to site 1 and the affinity of site 1 decreases with ligation. Finally, if 1 Xx - °XX = 0, then binding to site 2 has no effect on binding to site 1 and the two sites are independent. It should be pointed out that in this case the two sites are truly independent, since lXx — °XX = 0 only if c12 = 1. The binding capacity of site 1 is computed as in the global description by differentiating Xx or X2 with respect to InJC, i.e., = X,(l + lXx - X)
Bx =
(3.25)
dlnx By virtue of the definition of ^F in terms of °^; 1 and lxPl9 one also has
B, = (1 - XX){X - °X{) = X.il - Xx)(l + i * ! - 0X,)
(3.26)
Likewise for site 2 B2 = X2(l + *X2 -X)
= (l-
X2)(X - °X2)
= X2(l - X2)(l + *X2 - °X2) (3.27) The site-specific binding capacity has a peculiar feature, which makes it quite different from the analogous global quantity B. In fact, the sign of the binding capacity of an individual site, say site 1, is set by the sign of the expression 1 + 1X1 — °X1. We will see that, in general, this expression is not always positive. However, for the simple case of two sites, the sign of 1 + lXx — °Xi is always positive, since both lXx and °XX are bounded from zero to one. The expression 1 + lXx — °X1 is the same as the slope of the site-specific affinity plot, where l n * ^ is plotted versus lnx, plus one. We conclude from Section 2.2. that this slope must equal the site-specific Hill coefficient. In fact, the Hill coefficients of sites 1 and 2 are din nx =
\i-x din*
— = \ + iX i-
°XX
(3.28a)
din n2 =
— = 1+
X
X2 - °X2 (3.28b) dlnx The Hill coefficient in the local description is an important linkage
118
3 Local binding processes
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
oo
d
CO
d
d
q d
log x
3.1 The reference cycle
119
00
d
CO
d
CN
d
q d -3.0
-2.0
-1.0
1.0
0.0
2.0
3.0
log* Figure 3.4 Site-specific binding curve of site 1 for N = 2 for various cooperative cases: (a) effect of changing c\2 when K\ = 3 and # 2 = 0.33: c i2 = 10 (left), c12 = 1 (middle), c^ = 0.1 (right); (b) effect of changing K2 when # i = 3 and ci2 = 10: K2 = 3.3 (left), K2 = 0.33 (middle), #2 = 0.033 (right); (c) effect of changing K2 when # i = 3 and ci2 = 0.1: K2 = 3.3 (left), # 2 = 0.33 (middle), # 2 = 0.033 (right).
quantity, as will be shown in Sections 3.2 and 3.3. The difference l Xi — °Xl9 which is the slope of the affinity plot, gives the number of ligands exchanged at site 2 upon ligation of site 1. Hence, the Hill coefficient nx gives the net number of ligands exchanged upon ligation of site 1, including the ligand bound to this site. The response functions for site-specific effects dealt with so far are illustrated in Figures 3.4-3.7. The local and global descriptions are linked through a number of conservation relationships (Klotz and Hunston, 1975; Ackers, Shea and Smith, 1983; Di Cera, 1989). For the simple system of two sites dealt with above one has (3.29a) X= X2 B=
B2
(3.29b)
3 Local binding processes
120 d
CO
d
CN
d
q d -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
1.0
2.0
3.0
log x
CO
d
CN
d
q d
-3.0
-2.0
-1.0
0.0
log x
3.1 The reference cycle
q 6 -3.0
-2.0
-1.0
0.0
121
1.0
2.0
3.0
log x Figure 3.5 Binding capacity of site 1 for N = 2 for various cooperative cases: (a) effect of changing cu when K\ = 3 and #2 = 0.33: c i2 = 10 (left), cu = 1 (middle), ci2 = 0.1 (right); (b) effect of changing K2 when i^i = 3 and cu = 10: #2 = 3.3 (left), iC2 = 0.33 (middle), K2 = 0.033 (right); (c) effect of changing iC2 when Kx = 3 and ci2 = 0.1: X2 = 3.3 (left), K2 = 0.33 (middle), £ 2 = 0.033 (right).
X and B are extensive properties of the macromolecule when referred to those of the individual sites. Using the definition of *P in eq (3.1) one also has (Di Cera, 1989,1990) (3.30)
where j = 1, 2. The partial derivative indicates that all other variables in W are kept constant. The partial derivative 3/3 In Kj acts as a filter for the terms in W that are linear in Kj. From the definition of X = dln^/dlnx in the global description, the following operator representation holds dlnx
31n# 2
=V
(3.31)
122
3 Local binding processes
-3.0
-2.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
2.0
3.0
q
q 6
3.2 Contracted partition functions
123
q 6
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x Figure 3.6 Affinity function of site 1 for N = 2 for various cooperative cases: (a) effect of changing c i2 when Kx = 3 and # 2 = 0.33: c12 = 10 (top), cn = 1 (middle), c12 = 0.1 (bottom); (6) effect of changing K2 when Ki - 3 and c12 = 10: # 2 = 3.3 (left), K2 = 0.33 (middle), K2 = 0.033 (right); (c) effect of changing K2 when £x = 3 and ci2 = 0.1: # 2 = 3.3 (left), K2 = 0.33 (middle), £ 2 = 0.033 (right).
Here V is the 'gradient' operator associated with the system. The additivity of site-specific effects giving rise to the global picture should be appreciated in those cases outlined in Figure 3.2, where local and global cooperativity patterns do not coincide. A paradigmatic case is depicted in Figure 3.8. Positive coupling between sites 1 and 2 translates into negative cooperativity in the global picture due to site heterogeneity. The 'discrepancy' between local and global effects is particularly striking when examining the affinity plot. Although the affinity function of each site increases with ligation, that of the system as a whole decreases. 3.2 Contracted partition functions
In this section we deal with the concept of the contracted partition function (Di Cera, 1990), which plays a central role in our discussion of site-specific
124
3 Local binding processes
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
q
in
d
q d
3.2 Contracted partition functions
125
q
«* 2
in
d
o d -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x Figure 3.7 Hill coefficient of site 1 for N = 2 for various cooperative cases: (a) effect of changing c12 when Kx = 3 and i£2 = 0.33: c i2 = 10 (top), cn = 1 (middle), C12 = 0.1 (bottom); (6) effect of changing K2 when ATi = 3 and ci2 = 10: A:2 = 3.3 (left), K2 = 0.33 (middle), K2 = 0.033 (right); (c) effect of changing K2 when £ i = 3 and ci2 = 0.1: K2 = 3.3 (left), AT2 = 0.33 (middle), # 2 = 0.033 (right).
binding processes. We have already made use of these functions in previous sections (2.5 and 3.1), but without stressing their importance in general. Consider a macromolecule containing N binding sites. In the global description the partition function of such a system contains N + 1 terms, of which N are independent. In the local description, on the other hand, the ligation state of the N sites separately should be explicitly taken into account. Since each site can exist in two possible configurations, ligated or unligated, there are a total of 2N terms to be considered in the local description. Hence, the number of independent terms in the partition function of the local description is v = 2N - 1 (3.32) The operational complexity of the local description becomes fully evident at this point. While the number of independent terms in the partition
3 Local binding processes
126 LO
to O
6
IT)
9
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log x Figure 3.8 Affinity plot for N = 2 and site-specific parameter values A'I = 10, K2 = 0.1 and c i2 = 10. The affinity functions for site 1 (top) and 2 (bottom) are depicted by dashed lines and show positive microscopic cooperativity. The affinity function of the macromolecule as a whole is depicted by a continuous line and shows negative macroscopic cooperativity.
function of the global description grows linearly with N, the analogous number in the local description grows exponentially with N. It is convenient to consider each configuration as an Af-dimensional vector of binary digits, 0 and 1, that depict the unligated and ligated states of a site. For example, the vector § = [0011] (3.33) labels one of the sixteen possible configurations of a macromolecule containing four binding sites (N = 4), where sites 1 and 2 are unligated and sites 3 and 4 are ligated. In the general case of N sites f = [*j8 . .. a>] (3.34) with ar, /?, . . . w = 0, 1. The first index refers to site 1, the second index to
3.2 Contracted partition functions
127
site 2 and so on, up to the co index for site N. The partition function for this generalized ensemble at constant T and P is obtained by summing over all possible configurations, i.e., V(x) = 2 1 • • • 2 Y ^ . - * " + / 5 + - + t 0 or=0/3=0
(3-35>
co=0
The mechano-thermal partition function Yaj3... w is linked to the configuration labeled by the vector £ = [a/3. . .co] and to the Gibbs free energy G>0...e,= — A:BrinYa0...<»• Again, without loss of generality, we can set Yoo. .o = 1 and take the unligated form of the macromolecule § = [00 . . . 0] as reference. The standard free energy change for the reaction M00...0+ ( * + £ + . . . + a>)X&Map...t0
(3.36)
or equivalently, [00 . . . 0] + (a + /3 + . . . + co)X O [a/3 . . . co]
(3.37)
is given by
,,,ffl= GaP...m - G00...0 = ^B^ln
Yoo
---° = -fcBrinA^....,
(3.38)
As usual, we have cast all equilibria in terms of the overall equilibrium constants Aa^_w. The partition function (3.35) written in terms of these parameters is then
v w = 2 2 • • • 2^...«* flr+"+-+<0 a=0p=0
(3-39)
o)=0
The linkage among the N sites is uniquely defined in the site-specific description once the v = 2N - 1 independent coefficients A in eq (3.39) are known. We now cast the coefficients A in terms of site-specific binding constants K and interaction coefficients c as follows. The binding constant K; = Ax)...i...o (3.40) reflects the binding of X to site j of the macromolecule when all other sites are unligated. The value of Kj is associated with the standard free energy change AG, = -kBTlnYoo'A'"0
= -kBT\nKj
(3.41)
Yoo.. .0.. .0
The coefficients A for which # + / ? + . . . + c o = l define the site-specific binding constants K. The remaining coefficients define interaction constants for site-specific cooperativity. Consider the coefficient AQQ. .1. .1. .0
128
3 Local binding processes
associated with the configuration with all sites unligated, except sites i and 7. The interaction constant ctj defines the ratio Ctj =
Yoo.. .0.. .0.. .0Y00.. .1.. .1.. .0
^4oo.. .1.. .1.. .0
=
Yoo.. .1.. .0.. .0Y00.. .0.. .1.. .0
,~ Ars\ (3.42)
^ 0 0 . . .1.. .0.. .0^00.. .0.. .1.. .0
or the equilibrium constant for the dismutation Moo.. .1.. .0.. .0 + M o o . . .0.. .1.. .0 <=> M o o .. .0.. .0.. .0 + MQO. . .1.. .1.. .0
(3.43)
The interaction coefficient ctj establishes the stability of the doubly-ligated configuration involving sites i and 7 relative to the parent singly-ligated configurations. A value of ctj > 1 indicates that the ij-doubly-ligated intermediate is more stable than the sum of the parent singly-ligated configurations. A value of ctj < 1 indicates reduced stability and ctj = 1 indicates no difference. The standard free energy associated with cf/ is AG,y = — kBTInctj
= AG00...1...1...0 "" AG00...1.. .0...0 ~ AG00...0...1...0
= AG00...1...1...0-AG..-AG,
(3.44)
and gives the 'excess' free energy or driving force associated with the dismutation reaction (3.43). The definition of ctj allows the coefficient ^4oo.. .1.. .1.. . 0 t o be written in the simple form ^oo...i...i...o = c ^ ^
(3.45)
The recipe for defining interaction coefficients involving three or more sites is a straightforward generalization of the foregoing method. Consider the triply-ligated intermediate with sites /*, i and 7 ligated. Then . .1.. .1.. .1.. .0 =
c
hijKhKtKj
(3.46)
and chij is the equilibrium constant for the reaction Moo.. .1.. .0.. .0.. .0 + Moo.. .0.. .1.. .0.. .0 + MQO. . .0.. .0.. .1.. .0 <S> 2Moo.. .0.. .0.. .0.. .0 + Moo.. .1.. .1.. .1.. .0
(3.47)
In general, the coefficient Aa^mm(O of the partition function can be written as the product of an interaction constant c and a + f$ + . . . + co site-specific binding constants K. The index of c is assigned by listing the ligated sites of the intermediate in lexicographic order and, for each ligated site, the appropriate K is entered. For example, the product c2siK2K5K1 refers to the triply-ligated intermediate with sites 2, 5 and 7 bound. The interaction coefficient is a measure of the coupling among sites and the number of linked sites defines the order of the interaction. The zero-order coupling constant for the reference configuration and the first-order
3.2 Contracted partition functions
129
interaction constants for all singly-ligated configurations are identically equal to one, as follows directly from their definition. First-order interaction constants are unnecessary, since they would reflect the change in binding affinity of a given site relative to a reference state other than the macromolecule under consideration. The binding constant Kj encapsulates all the molecular events, tertiary and/or quaternary, linked to the binding to site j when all other sites are unligated. This constant is a property of the macromolecule under consideration and should not be expected to be the same as that of site j isolated from the rest of the macromolecule. The foregoing definitions of cs and Ks have the rather nontrivial consequence that all ligated intermediates are labeled in a linear fashion according to the ligated sites. Since each site is labeled as a separate entity in a unique way, it follows that each time site j is ligated the constant Kj must appear in the related coefficient of the partition function. On the other hand, if site j is unligated, then the related coefficient of the partition function does not contain Kj. Hence, although the partition function is a polynomial of degree TV in the ligand activity x, it behaves as a polynomial of first degree in Kj, and this is true for all j = 1, 2, . . . N. This suggests that W can be manipulated using the ^ s as independent variables in a rather simple way. It is at this point that contracted forms of W come into the picture. Consider a particular site, say site /, out of the total N sites. The probability of this site being ligated as a function of x is the site-specific binding curve Xj of site j . To compute Xj we proceed as follows. The whole ensemble of 2N configurations of the system is partitioned in two subsets: one containing all configurations with site j ligated and the other containing all configurations with site j unligated. Each subset constructed in this way contains a total of 2N~l terms. Each configuration belongs either to the first subset or to the second, and the information stored in each subset is complementary. Using the property that all configurations with site j ligated necessarily contain the term Kj, while those with site / unligated do not, the partition function (3.35) of the whole system can be written as W(x) = Ox¥j(x) + Ol
1}
¥j(x)KjX
(3.48)
F; is the partition function of the subsystem composed of N - 1 sites other than ; with site j kept unligated and lx F ; is the partition function of the subsystem composed of N - 1 sites other than j with site / ligated. These functions are contracted partition functions in the sense that they provide information on contracted forms of the macromolecule generated
130
3 Local binding processes
by keeping a given site in a particular ligation stated Like W, the contracted forms are polynomial expansions in the ligand activity x with positive coefficients. The various contractions involving the 2N configurations for N = 3 are shown in Figure 3.9. The function Xj is simply the ratio of all configurations with site j ligated to the total, i.e., Xl
_ K;iM
p.49)
. ! - !W
W(JC)
WO)
If we exploit Kj as an independent variable in eq (3.48), we also have (Di Cera, 1989, 1990) ^»^v^ (3 5 0 ) 3 In Kj which is a consequence of the fact that neither OxVj nor lx¥j are functions of Kj. The importance of eq (3.49) stems from the fact that the local quantity =
OOO
OOO
• OO
O«O
• •O
• €>•
O»O
OO»
«OO
#«O
OO» <
Figure 3.9 Manifold of ligated configurations for N = 3 (top). The ensemble can be split into two sets, one containing all intermediates with site / unligated (left), and the other containing all intermediates with site / ligated (right). The appropriate partition function is indicated for each case. The basic relationship (3.48) is readily derived from the figure. f
The term 'contracted' is introduced by analogy to the contraction of secular equations in quantum theory (Lowdin, 1964), which reduces the dimensionality of the space under consideration (Di Cera, 1990).
3.2 Contracted partition functions
131
Xj can be derived from knowledge of global quantities such as W and one of its contracted forms. In fact, W reflects the behavior of the system as a whole, while °Wj reflects the behavior of a contracted system with site j unligated. The contraction generates another macromolecule with N — 1 sites available for binding and site j frozen in either its unligated or ligated form. The contraction has absolutely no effect on the fundamental properties of either Ol F ; or ll F y that are analogous to those of W. More importantly, eq (3.49) poses no restrictions on the form of the coefficients of W and °\Py. Information on the site-specific quantity Xj can be arrived at from W and Oxf*j even if these coefficients are known in their global form. Hence, generating contracted forms of the system as a whole is equivalent to accessing information on the site-specific components involved in the partition function. This also implies that the analysis and resolution of site-specific energetics demands knowledge of the partition function of the system as a whole, as in the global description, and also of its contracted forms. Site-specific thermodynamics provides a connection between the properties of individual sites and those of the system as a whole and its contracted forms. Intuitively, we can anticipate that many site-specific properties can be cast in terms of global properties of the system as a whole and its contracted forms. The link originates from the properties of the contracted partition functions derived from W. Each contracted partition function in eq (3.48) can be written in a form analogous to W by properly choosing a reference configuration. The unligated form of the macromolecule is used as reference of W9 but in the case of contracted partition functions the choice may be different. Consider the case of ll F 7 . All configurations in 1W7- contain site j ligated and therefore it is appropriate to choose the configuration [00. . .1. . .0] as reference. In the case of °*Py, all configurations contain site / unligated and the configuration [00. . .0. . .0] may be taken as reference. Incidentally, this configuration is the same as the unligated form of the macromolecule. In either case, we note that the configuration taken as reference is the one with the maximum number of unligated sites compatible with the ensemble of ligated species. This simple rule applies quite generally even in the case of the system as a whole. Once the reference configuration has been selected, the properties of any contracted partition function can be treated as those discussed for the partition function of the system as a whole. For example, in the case of lx Uj9 the coefficients can be treated globally as overall equilibrium constants for the reaction (3.51) M;=1 + iX&Mj^Xi
132
3 Local binding processes
where M7=1 is the same as MQQ x 0? or the macromolecule with site j ligated and all other sites free. If A(/=i) is the equilibrium constant associated with (3.51), then (3.52) i=0
Note that the index i runs from zero to N — 1, since the contraction has reduced the number of sites available for binding. Likewise, for °V;- one has °Wj(x)= 2Ax,=o)*'
(3.53)
i=0
where -<4,(/=o) refers to the reaction analogous to (3.51) M H + iXoM/=0X, (3.54) Differentiation of eqs (3.52) and (3.53) yields all the response functions of the contracted macromolecule. Specifically, N-l
ftp
(3.55)
din N-l
N-l
1 V iA
•**i(j=l)%
vA
i=0
dlnx
A
2
(3.56)
VZ
\ i=0
I=0
1
N-l
2J i=0
1
( i V - l -
l
^i(j=l)x
(3.57)
i=0
are the average number of ligated sites, the binding capacity and the affinity function of the contracted macromolecule, respectively, with site ; frozen in its ligated configuration. Likewise, N-l
0Y
_
(3.58)
dlnx i=0
133
3.2 Contracted partition functions N-l
V i2A
d°Xj
N-l
x[
/ v^ ,• /i i=0
dlnx
(3.59)
N-l V A
1
v
i=0
\ i=0 N-l
(3.60) (AT - 1 -
Q A ^
1=0
are the average number of ligated sites, the binding capacity and the affinity function of the contracted macromolecule, respectively, with site j frozen in its unligated configuration. All these response functions obey the properties analyzed in Section 2.2. for analogous global quantities. In particular, the quantities %» (3.61a) x
m(j=0)
=
^N-l(j=0)
(3.61b)
give the mean ligand activities of the contracted forms of the macromolecule and provide a measure of the average work, in kBT units, spent in ligating one site. Site-specific response functions become clear when cast in terms of the foregoing quantities characterizing the behavior of contracted forms of the macromolecule. The affinity function of site j leads to the familiar form X =
****
(3.62)
1 + xKpc
Hence,
1}
xK = #
¥ (x) ^^
(3.63)
In general, the affinity function changes with saturation since site / is a subsystem subject to interactions with the remaining N — 1 sites. xKj is constant for all values of x only when l l P ; = °Wj.This case denotes the absence of linkage between site / and the rest of the macromolecule. If binding to site j does not affect the properties of the remaining N — 1 sites, then the two contracted partition functions are identical. The limiting values of the affinity function are of particular importance. For x —• 0 both contracted partition functions tend to one and therefore °Kj = Kj
(3.64)
134
3 Local binding processes
At low saturation the affinity function approaches the value of the site-specific binding constant Kj. For x -> °°, on theother hand, 1
"
~A
>
AN-1{j=o)
N-l
'
3 # 6 5
'
Xm(y=1)
The affinity function tends to a value that depends on the ratio of the mean ligand activities of the two contracted forms of the macromolecule. If binding to site y increases the overall affinity of the remaining N — 1 sites, then the mean ligand activity xm(;=i) is smaller than xm^=Oy Consequently, the affinity function increases with saturation and °°Kj>°Kj. On the other hand, if binding to site j decreases the overall affinity of the remaining N — 1 sites, then the mean ligand activity xm(;=1) is greater than *m(/=o) a n d °°Kj<°Kj. Hence, the behavior of the affinity function xKj of site y provides information on the behavior of the remainder of the sites of the macromolecule. This is even more evident when considering the slope of the affinity plot JJ (3.66) dlnx which gives a direct measure of the net number of ligands exchanged at the other N — 1 sites upon binding to site y. Site y is positively linked to the rest of the macromolecule ifJ AX > 0, since the number of ligated sites increases upon binding to site j . It is negatively linked if jAX < 0, since the number of ligated sites decreases upon binding to site y. In the case of j AX = 0, ligation of site y has no effect on the ligation of the remaining sites. The quantity jAX is the driving force for site-specific cooperativity. It is quite interesting to note that this driving force is provided by the sites of the macromolecule other than y. The response function xKj gives a measure of how sites communicate and give rise to cooperative interactions. The quantity jAX also defines the site-specific binding capacity. Differentiation of eq (3.62) yields by virtue of (3.66) Bj = Xj(l - Xj)(l + >AX) (3.67) The response function B] is the product of two terms. The term Xj(l - Xj) is always positive and gives the value of Bj when jAX = 0. This case corresponds to the absence of linkage between site j and the rest of the macromolecule, or a site j decoupled from the other sites. In this case site y behaves as a reference system and the binding capacity is given by the familiar form BOj = Xj(l - Xj)
(3.68)
3.2 Contracted partition functions
135
where the suffix on B indicates that site j is an independent site. The second term in eq (3.67) is the slope of the affinity plot in eq (3.66) plus one. This quantity, as we have seen in the global description, defines the Hill coefficient. The Hill coefficient of site / is therefore n7 = 1 + *AX = -^-
(3.69)
The definition in terms of the ratio Bj/BOj is analogous to that of the global description. The Hill coefficient can be seen as the ratio of the fluctuations of the quantity Xj in the system under consideration, relative to those observed for a reference system in which site j behaves independently. The term 1 + jAX, on the other hand, is peculiar to the local description and has no parallel in the global picture. This term is a linkage quantity which expresses the net number of sites ligated when site j is bound. In fact, jAX gives the net number of ligands exchanged at the other N - 1 sites upon binding to site ;. The unit term can be considered as the number of ligands exchanged at site j upon binding to this site. Hence (Di Cera, 1994b), Definition: The Hill coefficient of an individual site is the net number of ligands exchanged by the macromolecule upon binding to that site. If site j has n}? = 2.4, it means that binding to site j increases the number of ligated sites of the macromolecule by 2.4 units. Of this quantity, 1 refers to ligation of site /, while the remaining 1.4 units indicate the change in overall ligation of the remaining N — 1 sites. Unlike the term A}(1 - Xj)9 the term 1 + jAX in eq (3.67) can be positive or negative. The term jAX is the difference of two functions bounded from zero to N — 1 and is bounded from —{N — V) to N — 1. Hence, the site-specific Hill coefficient obeys the condition 2-N^n^N (3.70) For N = 1, this condition merely states that n}? = 1, as expected for a macromolecule containing a single site. For N = 2, the condition is analogous to that of the global case, with n7 being positive and bounded from zero to two. However, for N ^ 3, the lower bound for n} is no longer zero, but a negative number. The upper bound, on the other hand, is always the same as that of nH for the macromolecule as a whole. A negative value of the Hill coefficient, which would represent a violation of the second law of thermodynamics in the global picture, is perfectly legitimate in the local picture. The binding capacity, which is a positive
3 Local binding processes
136
quantity in the global picture, can be positive or negative in the local picture due to the definition (3.67). Negative values of the binding capacity have been documented experimentally (Kojima and Palmer, 1983; Hendler, Subba Reddy, Shrager and Caughey, 1986) and are shown in Figure 3.10. The physical significance of negative values of n; or B} should be rationalized in terms of the significance of y AZ. The peculiar meaning of rij in the local description reveals that this quantity is a measure of the net balance of ligation upon binding to site /. If binding to site j causes ligand X to be released at other sites due to negative linkage, then y A X < 0 . When jAX < - 1 the Hill coefficient of site j becomes negative and so is the site-specific binding capacity. We also note from eq (3.69) that rij —> 1 for x —»0 and x —» <», just as seen for the Hill coefficient
00
d
CO
d
CN
d
q d -0.5
-0.4
-0.3
-0.2
-0.1
0.0
(V) Figure 3.10 Electron binding curve of the heme center of cyta in the complex of cytaa3 (Kojima and Palmer, 1983). The redox potential provides the driving force for electron binding and scales linearly with the logarithm of the electron activity. The site-specific binding curve of cyta shows a negative binding capacity in the potential region of 200-260 mV.
3.2 Contracted partition functions
137
in the global description. This is because ;AJSf —• 0 in those limits. A final remark should be made on the extrema of the Hill coefficient n ;, which can be derived in a straightforward way from the properties of contracted forms of the macromolecule. The function rij has extrema for finite x whenever the derivative drij/dlnx vanishes, or when 1 Bj = °Bj (3.71) j This is also the condition for which AX has an extremum, i.e., it is a minimum or a maximum. In either case, the linkage between site j and the rest of the macromolecule, whether positive or negative, is maximized when the binding capacities of the two contracted forms with site / ligated and unligated become identical. Under this condition the cooperativity of site j is either a maximum or a minimum. Other properties of the contracted partition functions can be derived by recognizing that the definition (3.48) does not depend on the particular value of ;, so that for any two sites i and ; one has °^; (JC) + ^jWKjX = OxVi(x) + 1}¥i(x)KiX (3.72) Clearly, since there are TV sites, there must be N equivalent ways of casting *P in terms of contracted partition functions with one site frozen. However, each contracted partition function can in turn be split into contracted forms like W. For example, OlP7 can be written as 10 °Wj(x) = °°W ^(x)^x (3.73) iy(jc) + by contracting over site i once site / is kept in the unligated configuration. The contracted partition functions mxVtj and 1OlF/; with two sites frozen depict the behavior of the remaining N — 2 sites when site i and j are kept unligated, or when site / is unligated and site / is ligated. Likewise, for lx¥j one has ^ ( x ) = 01 ^ y (x) + ^WcqKpc (3.74) Hence, (3.75) W(x) = °°%.(x) +[10^.(jc)«i + °%(*)*;.]* + "VqWcyKiKpc2 N 2 Each of the contracted partition functions in eq (3.75) contains 2 ~ terms and the four terms in eq (3.75) yield a total of 2N configurations, as expected. There are N(N — l)/2 equivalent ways of casting W in a form like eq (3.75) in terms of contracted partition functions with two sites frozen. The contraction process can be extended up to N sites, in which case the contracted partition functions contain a single term equal to one. One can see by inspection that the partition function W can be written as the sum of 2m independent terms, each containing an mth-order contracted partition function with m sites frozen, times a 'weighting factor'
138
3 Local binding processes
reflecting the probability of existence of the reference configuration in the contracted manifold relative to the unligated form of the macromolecule, which is the reference configuration for the system as a whole. When W is cast as in (3.48), the whole system is analyzed in terms of site j under the influence of the remaining N — 1 sites. This influence, or 'perturbation', is embodied by the contracted partition functions lx¥j and Ox ¥j. The reciprocity of the perturbation also offers the alternative interpretation of the whole system in terms of the properties ofN — l sites perturbed by site j . The perturbation in this case is provided by the weighting factors 1 and Kpc. Likewise, when W is cast as in eq (3.75), the whole system is analyzed in terms of the interactions between sites i and j under the influence of the remaining N — 2 sites, as determined by ° ° ^ , 1Ox P/;, Olx¥ij and llxF,y, or alternatively in terms of the properties of these N — 2 sites under the influence of sites i and ;. The weighting factors in eq (3.75) are 1, Ktx, K-x and c^KiKpc2. Note that these terms are the elements of the partition function of a system containing sites i and j alone, just as the weighting factors 1 and Ktx in eq (3.48) are the elements of the partition function of a system containing site j alone. Hence, we have a rather simple rule for expressing W in terms of the properties of subsystems. The partition function of the system as a whole can be cast in terms of the partition function of any subsystem, provided each term of the polynomial expansion is complemented by an appropriate contracted partition function. For example, if we want to cast the properties of the system in terms of site /, we can write the partition function for a system containing site ; only, i.e., ^ ( J C ) = 1 + Kjx
(3.76)
and then correct each term by ad hoc contracted partition functions. The choice of which function enters the definition of W is determined by which configuration is represented by the terms in eq (3.76). The term 1 reflects the contribution of the unligated site y. Its complement is the contracted partition function of the remaining N - 1 sites with site / kept unligated. Likewise, the complement of Kpc is the contracted partition function of the remaining N — 1 sites with site j kept ligated, hence, eq (3.48). Analogous arguments lead to eq (3.75) using the partition function for sites i and / QW(x) = 1 + (Kt + Kj)x + CyKiKjX2
(3.77)
and the appropriate complements for each term, and so forth. One sees that in the limit ofm = N the partition function contains 2N weighting factors complemented by contracted partition functions all identically
3.3 Thermodynamic basis of site-specific effects
139
equal to one. This is the case where the expansion of W in terms of contracted partition functions yields the site-specific form of W. The other special case for m = 0 yields only one term, equal to one, complemented by a contracted partition function where none of the sites is frozen. The result is a mere tautology, since this zero-order contracted partition function is the same as W. 3.3 Thermodynamic basis of site-specific effects In the previous section we have illustrated some of the properties of site-specific response functions. In particular, we have seen that X] cannot be derived from the partition function W of the system as a whole by differentiation with respect to lnx, but only from contracted forms of W. We have also seen that Bj can assume negative values for N ^ 3. In this section we illustrate the thermodynamic origin of site-specific cooperativity and provide an explanation for the reason why site-specific effects are not necessarily subject to the restrictions imposed by thermodynamic stability (Di Cera, 1994b). Site-specific thermodynamics bears on fundamental features that are absent from the global description of binding phenomena dealt with in the previous chapter, as will soon be evident from consideration of the following multicomponent analogue. Consider the N sites of the macromolecule as each binding a different ligand and let X; be the ligand binding to site ;, with j = 1, 2, . . . N (see Figure 3.1). At constant T and P the relevant potential for this generalized ensemble containing N different ligands is dn = -kvT^Xjdlnxj
(3.78)
7= 1
where Xj is the amount of ligand X ; bound to site j and JC; is its activity. The partition function for this system is a straightforward generalization of eq (2.142)
W(xl9 x2, . . . xN) = ^ 2 . . . 2 Yap.mmmxfxZ #=0/3=0
...x»
(3.79)
co=Q
The first index refers to ligand Xl9 the second index to ligand X2 and so on. The similarity between eqs (3.79) and (3.35) is evident. The system of N ligands, each binding to a separate site, provides a multicomponent analogue of the local site-specific scenario where the same type of ligand binds to N sites of the macromolecule. Each ligation intermediate in the Af-dimensional manifold spanned by the N ligands is characterized by the
3 Local binding processes
140
Figure 3.11 Multicomponent analogue of site-specific binding phenomena, sketched as a macromolecule containing three sites, each binding a different ligand (squares to site 1, circles to site 2, triangles to site 3). Under conditions where all ligands are present at equal concentrations under any transformation the system at left is equivalent to a macromolecule containing three sites for a single type of ligand, as shown at right. mechano-thermal partition function Yapmtm(0 and the Gibbs free energy Gap...a = ~kBTlnYaj3 w. The standard free energy change for the reaction M + aX, + /3X2 + . . . + coXN O M(X0.(X 2 )^ . . . (X*)*
(3.80)
is given by A G 4 . , = GaP,mM - G00...0 = kBTIn
Y
° ° 0- = -kBTlnAapmm.<0
(3.81)
Yocfi. . ,o)
Again, we have recast all equilibria (3.80) in terms of the overall equilibrium constants Aapmmt(0, letting YQO...O = 1 without loss of generality. We recall that this is equivalent to selecting the unligated configuration of the macromolecule, Moo. ..o> a s reference. The partition function (3.79) is therefore (3.82)
x2, . . . xN) = (D=0
and contains a total of v = 2N- 1 independent coefficients, since there are a total of 2N possible configurations, of each one is used as reference. Knowledge of this set of coefficients completely defines the linkage scheme involving the TV ligands. The functions Xj and 5 ; can be derived from W by differentiation as follows A- =
(3.83)
3.3 Thermodynamic basis of site-specific effects
B, = ( 1 5 . )
141
(3.84)
where {x^ ; } denotes that all variables other than xy are held constant. It is easy to verify from eqs (3.83) and (3.84) that site / behaves noncooperatively as expected. The affinity function of site j can be derived by consideration of contracted forms of W written as follows
W({*,}) = °W«W) + M { W ) ^ * ;
(3-85)
The partition function of the system as a whole can be expressed as the sum of two terms, one containing all ligated configurations with site j unligated, °^/ ({x^ ; }), and the other containing all configurations with site j ligated, lxf*({xi¥s j})KpCj. The coefficient Kj = A00...1...0 is the equilibrium constant for ligand X ; binding to site j when all other sites are in the unligated form. Note that the contracted partition functions Ox¥ and lxU are functions of all xs but x;-. Since there are N sites, then there are TV equivalent ways of casting W in a form given by eq (3.85), one for each binding site. The binding isotherm Xj cast in terms of contracted partition functions is X>
= j^5!«3s» = 1 - *««"»
(3.86)
It should be pointed out that the form of eq (3.86), which is of course equivalent to eq (3.83), follows directly from the definition of contracted partition functions and does not involve differentiation of V. The foregoing definition of Xj allows one to write the affinity function of site j as */}> (3.87) ) As expected, the affinity function is a constant, since it depends on all xs but Xj. Application of arguments developed in Section 2.5, and particularly eq (2.121), yields ly
Oy
i\
V
C\
QQ\
The change of the logarithm of the affinity function of ligand X ;, due to a change of the logarithm of the activity of ligand X/9 gives the net amount of ligands X, exchanged upon binding of ligand X ;. The quantity ]'AXt is specific of X,- and X;, and its sign defines the nature of the linkage, positive or negative, between ligands X,- and X;. The basic properties of the foregoing linkage scheme can be defined in
142
3 Local binding processes
terms of response functions computed when a given set of activities are kept constant. However, thermodynamics puts no restriction on what particular values this set of activities must satisfy. Consider, therefore, the case in which all xs are identical under any transformation. This is equivalent to a system of N ligands, which are present in solution at the same activity and bind to N distinct sites of the macromolecule. Any transformation that changes the activity of a given ligand also affects the activity of all others to the same extent, so that Xi = x2 = . . . = xN = x (3.89) Under this constraint, our original linkage scheme involving N different ligands binding to N distinct sites of the macromolecule becomes equivalent to a system composed of a macromolecule containing N sites for a ligand X, whose activity is x. In fact, the partition function (3.82) under the constraint (3.89) becomes V(x) = 1 2 • • • 2 A«p.. . mx^+-• or=0)3=0
-+°>
(3.90)
(O=0
The indices or, (3, . . . co labeling different ligands in the linkage scheme, now label different sites: site 1 for ligand 1, site 2 for ligand 2, and so forth. The partition function (3.90) is mathematically identical to eq (3.39). The sum ; = ff+j8+... + ft> (3.91) runs from zero to N, as expected for the partition function of a macromolecule containing N sites for a ligand X. Also, for each value of y, there are as many as CjjN coefficients Aapmmt0) in (3.90) that contribute to the overall coefficient of xJ in the polynomial expansion. The analogy with the linkage scheme involving different ligands makes it possible to recognize the site-specific contribution to the global description. The conceptual basis of this analogy can be summarized with the equivalence ligands (global description) = sites (local description) (3.92) The properties of individual sites of biological macromolecules can thus be derived from the properties of a multicomponent analogue composed of different ligands, each binding to a distinct site of the macromolecule, when the activities of all ligands are constrained to be the same under any transformation. Consider again the linkage scheme involving N ligands, under the constraint (3.89). When eq (3.89) holds, it is not possible to derive the amount of ligand X; bound to the macromolecule by differentiation of the
3.3 Thermodynamic basis of site-specific effects
143
partition function W with respect to lnx 7. The partial differentiation in eq (3.83) implies that all xs other than x7 are held constant. This condition, however, is at variance with the constraint (3.89), which establishes that all xs are the same under any transformation. A change in x7 necessarily leads to a change in all other xs. Under this condition, only the total derivative dlnW/dlnx, can be computed. The differential operator for a function of N variables is given by d In x7
rlin 3^ \\ dlnx,v /=i \ 3 In xt J {w d In x7
(3.93)
Under the constraint (3.89) all derivatives at the right-hand side become identically equal to one, while d/dlnx 7 is the same as the derivative d/dlnx. Hence, A d
dlnx
N
- WI
^ s
\
|
,-=i\ 3 In *,•/{*,*,•}
(3.94)
The action of the gradient operator V is equivalent to a total differentiation with respect to lnx, or to the sum of partial differentiations with respect to all variables x in the usual way. Hence, the properties of the linkage scheme under the constraint (3.89) can be derived from those of the same system in the absence of constraint. Partial differentiation of the partition function (3.85) with respect to lnx7, under the constraint (3.89), is equivalent to application of the V operator to In W, which yields (3.95) 7= 1
The presence of the constraint yields a quite different result. Instead of the function Xj, reflecting the amount of ligand X7 bound to the macromolecule, the differentiation yields the sum of the Xs for all ligands present in solution, or the total number of ligands bound to the macromolecule. This provides a simple explanation for the impossibility of deriving site-specific binding isotherms from differentiation of the partition function W with respect to lnx, when dealing with the binding of ligand X to N sites. The quantity Xj in the multicomponent analogue, in the presence of the constraint (3.89), acquires a site-specific character and needs to be derived in a different way. Eq (3.86) proves extremely useful in this regard. Knowledge of the partition function and its contracted forms in (3.85) provides a definition for Xj regardless of the particular conditions. Once Xj is obtained, other important response functions can
144
3 Local binding processes
be derived from it. The binding capacity of ligand X ; is defined as the derivative of Xj with respect to lnx ; . Again, if eq (3.89) holds, only the total derivative dXj/dlnXj can be computed. Application of the operator relationship (3.94) yields. VXj = Bj + Lyl + Ly2 + . . . + LjN
(3.96)
The change of Xj with lnx ; under the constraint (3.89) does not give the binding capacity in the absence of the constraint, Bj, but the sum of Bj and all other linkage coefficients Ljt (i =£ ; ) . Switching to the case of individual sites using the equivalence (3.92) and the significance of the gradient operator V, it results that the binding capacity of an individual site, dXj/d In x, contains the contribution of the linkage between that site and the rest of the macromolecule. The important consequence of eq (3.96) is that the second law of thermodynamics, which applies to Bj, has no control on VXj in the case of different ligands under the constraint (3.89). Hence, the constraints imposed by the second law do not apply in the case of the binding capacity of an individual site and eq (3.96) provides the thermodynamic basis of site-specific effects. The sign of the expression (3.96) can be positive or negative, depending upon the relative contribution of the linkage coefficients that can themselves be positive or negative. Another important property of the multicomponent analogue under the constraint (3.89) is demonstrated by the affinity function (3.87). When the logarithm of the affinity function is changed with respect to lnx ; in the presence of the constraint, the gradient operator applies and yields the result + >AX2 + . . . + 'AXN
(3.97)
where the various terms on the right-hand side have the same significance as in eq (3.88), and represent the amounts of ligands X l9 X 2 , . . . XN exchanged upon binding of ligand j . In the absence of the constraint, the change in the affinity function with Xj is equal to zero. No such result is guaranteed in the presence of the constraint. The Hill coefficient of ligand Xj is obtained from eq (2.44) by adding one to eq (3.97), i.e., rij = 1 + jAXx + jAX2 + . . . + jAXN
(3.98)
The value of n7, which is always unity in the absence of constraint, can be positive or negative depending on the value of the individual jAX^. The upper and lower bounds for eqs (3.97) and (3.98) can easily be calculated by recalling the definition of jAX{ in eq (3.88). For the linkage scheme under consideration, jAXt is bounded from - 1 to 1, since this quantity expresses the difference between two binding curves that change from
3.3 Thermodynamic basis of site-specific effects
145
zero to one. Since there are N — \ such terms in the sum in eq (3.97), then the right-hand side must be bound from — (N — 1) to TV — 1. Hence, the Hill coefficient of ligand X ; , in the presence of the constraint, is bounded from 2 — N to N. This is essentially the result derived for the Hill coefficient of an individual site in eq (3.70), and eq (3.98) provides the thermodynamic basis for site-specific cooperativity. The linkage with other sites generates cooperativity in site-specific binding curves. Eqs (3.70) and (3.98) give the quantitative expressions for this effect. The constraint (3.89) also reduces the dimensionality of the system. In a system containing N different ligands, each binding to a distinct site of the macromolecule, the potential (3.78) is associated with an TV-dimensional matrix G' with coefficients
ri - (^M \9 I /
{
}
0.99)
The matrix is positive-definite, as already demonstrated in Section 1.3. In the presence of the constraint (3.89), the potential (3.78) becomes N
dn = -kBT^Xjd\nx
= -kBTX dlnx
(3.100)
7= 1
where X denotes the total number of ligands bound to the macromolecule. The constraint has acted upon the system by contracting the dimensionality from N to one. Although N ligands are present, their activities do not change independently and are always equal, regardless of the transformation. As a result, the response function of the potential to a change in the independent variable lnx gives information on the overall quantity X, and not on the individual quantities Xj. Consequently, the derivative dX/dlnx is always positive due to thermodynamic stability. This derivative is equivalent to N
VX = V^Xj>0
(3.101)
7= 1
by virtue of (3.94). Consideration of eqs (3.96) and (3.99) yields (3'102) The action of V on Xj is equivalent to summing all metric coefficients of the yth row of G'. We recall that the symmetric nature of G' makes this result equivalent to the sum of all metric coefficients of the yth column. Hence, VX = dX/dlnx is the sum of all coefficients of G' and is always positive, since G' is positive-definite.
146
3 Local binding processes
3.4 Properties of the binding curve in the local description The properties of site-specific binding isotherms are somewhat analogous to those discussed in the case of the global description. We start by noting the conservation relationship X = J jX i
(3.103)
7= 1
The sum of the probabilities of binding to each site separately gives the amount of ligand bound to the macromolecule. The global property X is derived as the sum of all analogous local quantities. Since the quantity X is the derivative of In W with respect to In x, and each Xj can be derived as in eq (3.50), it follows that — = E^— =V (3.104) dlnx i=i 3 In A^Differentiation with respect to \nx is equivalent to the sum of partial differentials with respect to the variables K. This is a consequence of the fact that W is a linear function of Kt and that each K( labels the corresponding site unambiguously. The conservation relationship (3.103) has some important consequences in the analysis of site-specific effects. The mean ligand activity of the macromolecule in the global description is given by the integral equation (2.46) (3.105) NJo We have already pointed out that this quantity expresses the mean value of the ligand chemical potential in kBT units, or the average work spent on ligating one site of the system. Upon substituting X in eq (3.105) with eq (3.103), we obtain a sum of N terms of the form „•= flnxdXj f'lnxdX (3.106) Jo The quantity lnx m; is the logarithm of the mean ligand activity of site /, or the work spent to ligate site / in kBT units. Hence (Ackers et al.9 1983; Di Cera, 1989), 1 N lnjcm = - 5 > * m / Nj=i
(3.107)
which yields another conservation relationship. The mean ligand activity of the macromolecule is the sum of the work spent to ligate each site separately, divided by the number of sites.
3.4 Properties of the binding curve
147
The value of \nxmj is the value of lnx where the area under the Xj curve, <2u(x), equals the area above the Xj curve up to the asymptote Xj? = 1, Qa(x), this property being equivalent to that of lnx m in the global description. Likewise, the value of lnjcm/ can be derived from numerical analysis of the binding isotherm Xj from any point along the In x axis using the same method as described in Section 2.3 for the function X. Specifically, one has (nXXjd\nxf
Qu(x)=
(3.108a)
J — 00
Q&(x) = f°° (1 - Xj)dlnx'
(3.108b)
Jin x
Hence,
lnx m / = hue + [Qa(x) - Qu(x)]
(3.109)
for any value of x. Given the arbitrary point lnx l 9 the value of <2u(*i) is given by the area under the curve Xj from -oo to lnxi and likewise the value of <2a(*i) is given by the area above Xj up to the asymptote Xj? = 1, from In*! to oo. The difference GaC^i) ~~ Qu(xi) added to lnx x gives In x my , regardless of the particular point xx. The analytical solution of the integral in eq (3.106) for the mean ligand activity of an individual site is not as simple as in the case of the global description, since the function Xj cannot be obtained from differentiation of the partition function W with respect to In x. The difficulty is immediately evident if W is cast as in eq (2.78), i.e., N
(3.110) where or,- is the ith coefficient of the factorization related to the ith root of *P. From (3.50) one has
ill + o v, = ^ L f H 7
(3.111b)
31nK
Except for the coefficients v, Z ; has a form identical to X in the global description (see eq (2.79)). These coefficients can be real or complex conjugate, depending on the nature of the as with which they are associated. They can be thought of as 'filters' that distort the shape of the overall binding curve X to yield Xj. Some properties of these coefficients
148
3 Local binding processes
can be derived in a straightforward manner from consideration of the conservation relationship (3.103). We know from eq (2.79) that (3 n 2 )
The sum of all terms as in eq (3.111a) must give eq (3.112), because of eq (3.103). Hence, 2>0 = 1
(3.113)
7= 1
It also follows from eq (3.111a) that 5>v = 1
(3.H4)
i=i
since Xj —» 1 when x —> °°. The vs belong to an N x N matrix for which the elements of each row and column add up to one. The exact form of these coefficients depends on the form of the as written in terms of the site-specific variables K and c. Without an explicit form for these coefficients, solution of eq (3.106) is not straightforward. An intriguing consequence of eq (3.111a) is that the site-specific binding curve Xj can be expressed as the derivative of an ad hoc function with respect to lnx as follows
with N
Wj(x)
= ;=i
It follows from the properties of the vs that
W(x) = ^(x)W2(x) . . . WN(x)
(3.117)
The WjS are pseudo partition functions for the individual sites. Each Wj resembles W of the system as a whole, except for the filtering effect of the vs. Analysis of the shape of site-specific binding isotherms reveals other interesting aspects of the local description. When all sites are independent of one another, all coefficients c in the partition function W are equal to one and W assumes the rather simple form W(x) = [1(1 + Kpc) 7=1
(3.118)
3.4 Properties of the binding curve
149
In this case the coefficients a are real and coincide with the binding affinities of the individual sites. Also, the matrix of the coefficients v degenerates into the identity matrix, since vtj = 1 if i = j and v,y = 0 otherwise. Under these conditions, the site-specific binding curve is ^jX (3.119) ; 1 + Kj and is always symmetric around the center of symmetry xsj = Kj1. If all sites are not only independent but also identical, then K1 = K2 = . . . = KN= K and the system as a whole becomes a reference system. The site-specific binding curve of any site is merely X/N and is therefore symmetric like the global binding curve around xs = xsj? = K~x. In general, the symmetry properties of X have no bearing on those of Xj and vice versa (Di Cera et al.9 1992). The sum of symmetric functions is not necessarily symmetric and conversely the sum of asymmetric functions is not necessarily asymmetric. Consideration of the symmetry properties of Xj is, however, of great practical utility due to the difficulty of deriving analytical expressions for xmj. It should be recalled that for a symmetric binding curve the mean ligand activity is the same as the center of symmetry, or the value of x at half saturation. In order to investigate the symmetry properties of A}, we recall a result derived in Section 2.3 which states that if the binding curve X is symmetric, then the logarithm of the affinity function, \nxK, is also symmetric with the same center of symmetry. This result applies to Xj as well, since the affinity function XK is a simple transformation of X. From the site-specific affinity function in eq (3.63), and the condition of symmetry for ln*^ in eq (2.63), we obtain the condition for the site-specific binding curve Xj to be symmetric as follows Xj
=
v-) = ^
xSjX-i)
where xsj is the center of symmetry at half saturation, or else the mean ligand activity xmj of site /, and the condition must hold for any A ^ 0. The condition (3.120) involves the contracted partition functions Ox¥j and lx P ; , but imposes no constraints on the symmetry properties of the contracted forms of the macromolecule. In fact, symmetry of Xj does not require symmetry of Ox¥j and ly¥j. The explicit analytical condition for symmetry of Xj involves the coefficients of °Wj and l l F ; , along with Kj. We now rewrite °Wjand l l F ; in eqs (3.52) and (3.53) in the more convenient form l
Wj(x)= £/?,*< i=0
(3.121a)
150
3 Local binding processes
Ol
W = Sr*'
(3-121b)
i=0
with j80 = y0 = 1. Substitution of eqs (3.121a) and (3.121b) into (3.120) yields N-lN-l
2 E (AA> - YiYh«t)xTX- h = °
/*=0 i=0
( 3 - 122 )
Since eq (3.122) must hold for any A ^ 0, one necessarily has N-lN-l
E E (A/8* - y.-y*^)*:;* = °
h=0 i=0
(3-123)
for any i and /i such that i — h is constant. The relationships above are invariant upon the substitution i <•> h and therefore only half of them are truly independent. Letting i — h = n in eq (3.123), and considering only the independent relationships yields X "(&+„& - Yh+nYh^)xl1+n = 0
(3.124)
/l=0
There are a total of N such conditions, corresponding to the different values o f n = 0 , 1,2, . . . N — 1. Each condition for a given value of n is only necessary, but becomes also sufficient when the other conditions hold. On the other hand, if Xj is symmetric, then all the conditions (3.124) necessarily apply. One is particularly important and is obtained for n = N - 1. This gives fiN-i - OJ^YN-I = 0, or (3i25)
which provides a simple analytical expression for the center of symmetry of the curve Xj. This value coincides with the mean ligand activity of site j . It is worth pointing out that eq (3.125) is a direct consequence of the symmetry of \nxKj. In fact, eq (2.63) implies Xn^Kj + l n ^ " 1 ^ = -21njtsy (3.126) or equivalently, x^Kf^Kj = x~2 (3.127) Since eq (3.127) must hold for any A ^ 0, letting A = 0 yields *Ki~Kj = xJ (3.128) We know from eq (3.64) that °Kj = K} and from eq (3.65) that X/YN-I, which substituted into eq (3.128) yield eq (3.125).
3.4 Properties of the binding curve
151
We can now explicitly solve the problem of symmetry for local binding processes using two simple examples. In Section 2.3 we have demonstrated that X is always symmetric in the global description for N = 2. However, this condition does not apply in general to Xx and X2. Consider the case of Xx and the contracted partition functions ^ ( x ) = 1 + K2x (3.129a) 1 ^F1(x) = 1 + c12K2x (3.129b) Here yx = K2 and ft = c12K2, and the conditions (3.124) are given by ft - 7 l C ^ = 0 (n = 1) (3.130a)
1 - a? + (0? - yW)xi = 0
(n = 0)
(3.130b)
These conditions are satisfied only if yx = ft, which implies c12 = 1, or if YiPixli = 1? which implies Kx = K2. This means that Xx is symmetric only if the two sites are identical (Kt = K2) or independent (c12 = 1). The former case corresponds to Xx = X2 = X/2, and the latter to independent sites. Hence, for N = 2 a site-specific binding curve is always asymmetric, unless the two sites are identical or independent. For TV = 3, the global description yields a binding curve that is symmetric if A3 = (A^AJ3. The partition function of the macromolecule as a whole is W(x) = 1 + {Kx + K2 + K3)x + (c12KxK2 + CuK.K, + c23K2K3)x2 + cX23KxK2K3x3 (3.131) and the contracted partition functions for Xx are °^x(x) = 1 + (K2 + # 3)jt + c23K2K3x2 (3.132a) 2 ^ ( x ) = 1 + (c12K2 + c13K3)x + c123K2K3x (3.132b) Here 7 l = /^2 + #3, y2 = c23K2K3, ft = c 12 # 2 4- c13K3, ft = c123K2K3 and the conditions (3.124) are given by ft _ y 2 ^ = 0 (n = 2) (3.133a) A - 7 1 ^ + (Aft - YiYiO?)xl = 0 (/i = l) 1 - a) + (£1 - A^)xs! + ( $ - y ^ ^ s i = 0 (ft = 0) Symmetry of Xx demands /32y\ = y20l and K\fix = 71 y2, or 1
c 123 (# 2 + K3)2 = c23(c12K2 + c13K3)2
(3.133b) (3.133c) (3.134)
and K\{c12K2 + c13K3) = c23K2K3(K2 + K3) (3.135) It is possible for a site-specific binding curve to be symmetric for N = 3. The trivial case where all cs are equal to one, which again corresponds to
152
3 Local binding processes
independent sites, is embodied by the conditions (3.134) and (3.135) as a special case. However, symmetry in the global description does not necessarily imply symmetry in the local description. We have seen that for N = 2 the global binding isotherm is always symmetric, while the sitespecific isotherms are always asymmetric, except in a few special cases. For TV = 3, the conditions for symmetry in the global description are completely decoupled from those applying in the local description. Of particular interest is the simple case where all interaction constants are equal to one. In this case, X is the sum of three symmetric site-specific curves, Xl9 X2 and X3y but is itself symmetric only if one of the Ks is the geometric mean of the other two, e.g., if K\ = K2K3. When all Ks are identical, X is symmetric when 27c123 = (c12 + c13 + C23)3, and Xx is symmetric when c123 = c23, provided 2c23 = c12 + c13. 3.5 Properties of the binding capacity in the local description In Sections 3.2 and 3.3 we have demonstrated that the binding capacity of an individual site behaves quite differently from the analogous quantity reflecting the properties of the macromolecule as a whole. For a given site, 7, Bj can be positive or negative if N ^ 3. This is a consequence of the significance of the Hill coefficient and the affinity function in the local picture. Using the definition of X] in eq (3.49), the binding capacity of site j can be written in a number of equivalent forms as follows B; = Xj(l - Xj)(l + lX} - °Xj) = X j i l + 1 X y - X ) = (l-
Xj)(X - ° Xy )
(3.136) In all these forms, Bj is expressed as the product of two terms, one always positive and the other of variable sign if N ^ 3. We also have the conservation relationship analogous to eq (3.103) B = ^Bj
(3.137)
7=1
Since B is always positive, as a consequence of the second law of thermodynamics, negative values of the binding capacity of one site must be compensated by positive values at other sites to keep the sum (3.137) positive. Hence, from eq (3.137) we derive the rather obvious result that it is not possible for a macromolecule to have all sites showing negative values of the binding capacity. Given the fact that Bj can in general be positive or negative for N ^ 3, the question arises as to whether we can use Bj as a probability density function for the variable In x in a way analogous to B.
3.5 Properties of the binding capacity
153
By definition, a probability density function must be positive everywhere in the domain of definition of the independent variable (Feller, 1950). We note, however, that Bj = 0 for x -» 0 and x --> oo and that the integral C Bjdlnx = 1
(3.138)
J — 00
exists and is finite, regardless of the properties of Bj. This fact allows us to define a moment generating function for site j as follows
= fx^Bjdx = fx Jo
(3.139)
Jo
The function exists because of eq (3.138). We note from (3.139) and (3.137), that the sum of the moment generating functions of the individual sites gives the moment generating function G of the macromolecule as a whole, i.e., )
(3.140)
Substitution of Xj in eq (3.139) with the expression given in eq (3.111a) gives 2
W
oi=i(l + a(x)2
dx
(3.141)
and again convergence of the integral necessarily demands 0 < e o < l . Solution of the integral (3.141) is straightforward using the treatment given in Section 2.4 and yields ) = -J^-^vijar
(3.142)
which is analogous to G, except for the presence of the coefficients v. The conservation relationship (3.140) follows immediately from (3.142) because of the properties of the vs embodied by (3.113). The first two moments of the Bj distribution are derived from the Taylor expansion of Gj. As in the case of G, we let G; = fh with f(co) = -JI^Li
h(co) = £ v ^ r
(3.143a)
(3.143b)
154
3 Local binding processes
The Taylor expansion of / up to second order is derived from eq (2.90) as /(a,) = i + U!L 6 The Taylor expansion of h up to second order is
h(a>) = 1 - hvijlna\w+
/ Hence, the first two moments of Bj are
(3.144)
U^u**2 «)<>? 2\
(3.145)
/
N
(Inxj) = — ^V/ylnor,N
2
2
(3.146a)
(i n x 7 ) = — + ^vtjIn 2 af
(3.146b)
The first moment equals lnx m; . The second moment is proportional to the sum of squares of the logarithm of each coefficient a, multiplied by the appropriate coefficient v. The site-specific cooperativity associated with site ; is again measured by the dispersion of the Bj distribution along the lnx axis, or the variance o)=
ff =
i
-
* N
+
N
- H S vuvhj In2 —
3
2;-i*-i
(3.148)
<xh
Upon expressing each coefficient a in polar coordinates in the complex plane as a = r expi
^^
—j
3
_ ^
^
'Vv'Vu' I n
2i=\h=i
— -^j^jVijVhjKVi
™i 1 x
rh ~
x
V / ' V i . * ! CDi ™"
i=ih=\ (3.149)
As in the case of the global description, the mathematical 'driving forces' for site-specific cooperativity can be understood from the expression above involving the variance of Bj. There are four components that determine the value of o). The first component is the variance of the reference system, o\. The second component, a< r >, is the variance of the r values, filtered by a proper convolution of the vs. In fact,
3.5 Properties of the binding capacity
^ 2 2 > / / V n 2 - = lvhjln2rh - hv^lnrX 2i=\h=i
rh
h=\
\h=\
155
1
= o\r) (3.150)
Likewise, the third component, d\
= 2 w 2 * - h vhj(ph)2 = 2{(p) o h=l
\h = l
(
I
The fourth term is most interesting, as it is peculiar of the local description and does not appear in the definition of a2 in the global description. This term represents the covariance, o\r(p), of the joint distribution of the r and cp values, convoluted with the appropriate coefficients v. In fact, N
N
N
*Z E yvyhj(
Th
h=l N
N
2 i ^ yhj
(3.152)
h=l
The covariance so defined is a complex number, since it is multiplied by the imaginary unit. Hence, o2j = o20 + o\r) - a2{(p) + 2io\rq)) (3.153) It is straightforward to prove that the covariance is the result of the differences between the values of the vs. If v,y is the same for all i = 1, 2, . . . N, then o\rq)) = 0 because 5>* =0 ^cphlnrh
=0
(3.154a) (3.154b)
h=i
The other terms in eq (3.153) remain finite. The various terms in eq (3.149) oppose each other in defining the value of a2. Although the analysis of the contribution of each term is complicated by the presence of the coefficients v, qualitatively the variance of the norm of the coefficients or, o\r), tends to increase the value of o) with respect to o\, while the variance of the angles formed by the as in the complex plane, o2^, tends to decrease the value of o2 with respect to a§. In addition, the covariance term o\r(p) makes a contribution to the sum whose sign cannot be easily predicted.
156
3 Local binding processes
A final point should be made on the connection between the statistical moments in the global and local descriptions. The first two moments of the B/N distribution of the macromolecule as a whole are 1
N
2
i
(3.155a)
N
(In x) = — + — Y In2 a* (3.155b) 3 N/Ti as shown in Section 2.4. The definitions of vhj and ah are such that Vh.
= * ^ * = lhl!L + i_M_
dlnKt 31nA:y 3 In Kj Hence, the first moment in eq (3.146a) can be rewritten as
(3.156)
(3157)
The term in the summation is related to the second moment of the B/N distribution in eq (3.155b), and specifically ln2^> 1
(3.158)
2 a in*;.
The derivative of the second moment of the B/N distribution with respect to In Kj gives the first moment of the 5 ; distribution, times a factor that depends on the number of sites. Since (ln;c ; ) is the same as the mean ligand activity of site y, lnjcm;, and the sum of such terms is related to the mean ligand activity of the macromolecule by virtue of eq (3.107), summing eq (3.158) over all values of j yields the interesting result lnjcm = = --V(ln 2 jt> (3.159) 2 The mean ligand activity of the macromolecule as a whole can be derived from the second moment of the B/N distribution through the gradient operator. This operator acts as a filter on the second moment. The importance of eq (3.159) stems from the fact that the first two moments of the B/N distribution in the global description are not truly independent. The connection, however, can only be seen from consideration of the local properties of the macromolecule and cannot be drawn in terms of the global properties. The gradient operator provides the filtering function through which the first two moments of B/N can be obtained from one another. It also follows from eq (3.155a) that Vlnx m = - 1 (3.160)
3.5 Properties of the binding capacity
157
due to the properties of the coefficients v. Hence, from the definition of the variance in the global description and eq (3.159) it follows that Va2 = 0 (3.161) a rather intriguing result. The variance of the global binding capacity is independent of the gradient defined by the site-specific binding constants.
4 Specific cases and applications
In this chapter we present a systematic treatment of site-specific effects in systems composed of a small number of sites. We also deal with applications of the concepts illustrated in previous chapters to the analysis of cooperative systems of biological relevance.
4.1 ThecaseiV = 2 The simplest cooperative system is provided by a macromolecule containing two sites. Although some of the properties of this system have already been discussed in Section 3.1, here we give a systematic treatment for the sake of clarity and completeness. The partition function for the case N = 2 can be written in either of the following forms W(x) = 1 + Axx + A2x2 W(x) = 1 + (KX + K2)x + cnKxK2x2
(4.1a) (4.1b)
The first is the global form of \P, while the second is the local form. The coefficients Ax and A2 are the overall equilibrium constants in the global description, while Ku K2 and c12 are the relevant site-specific parameters. From the definition of W it follows that A1 = K1 + K2 A 2 = cX2KxK2
(4.2a) (4.2b)
and therefore the site-specific parameters cannot be assigned uniquely from knowledge of the global coefficients A. There are a total of four distinct configurations in the local description, corresponding to the four terms of the partition function (4.1b). These configurations can be associated with the two-dimensional vectors 158
4.1 The case N = 2
159
[00] [10]
[01]
(4.3)
[11] The unligated form of the macromolecule, M, is the same as the configuration Moo labeled in a site-specific fashion. This form is used as reference in the definition of the partition functions (4.1a) and (4.1b). The singly-ligated form of the macromolecule, MX, is the sum of the concentrations of the site-specific configurations M10 and MOi, corresponding to the intermediates with site 1 and site 2 ligated respectively. The doublyligated form, MX2, is the same as M n . Since there are four configurations in eq (4.1b), of which one is used as reference, then there must be three independent reactions among the possible intermediates leading to a total of three independent equilibrium constants. These are the three site-specific parameters K u K2 and c12 that represent the equilibrium constants for the reactions MQO + X o Mio
(4.4a)
Moo+ X ^ Moi
(4.4b)
M10 + Moi <> MQO + Mn
(4.4c)
In the first reaction site 1 is ligated while site 2 is kept unligated, and the equilibrium constant is Kx. Likewise, in the second reaction site 2 is ligated while site 1 is kept unligated, and the equilibrium constant is K 2. The third reaction is a dismutation where the unligated and doubly-ligated intermediates are generated from the two singly-ligated configurations. The equilibrium constant for this reaction is c12 and is a measure of the linkage between the sites. The parameters Ax and A2 of the global description refer to the reactions M + XoMX + 2XoMX2
(4.5a) (4.5b)
We also have, from arguments discussed in Section 2.2, two stepwise binding constants kx and k2 for the reactions M + X<^MX
(4.6a)
MX + X<>MX 2
(4.6b)
with At = 2kt
(4.7a)
A2 = k xk 2
(4.7b)
160
4 Specific cases and applications
and conversely fci = — 2
(4.8a)
k2 = ^
(4.8b)
In the absence of interactions between the sites, c12 = 1 and the partition function assumes the rather simple form W(x) = (1 + KlX)(l + K2x) (4.9) Each site behaves as an independent unit which provides a binomial factor to*P. We have seen in Section 3.1 that the case c12 = 1 does not correspond to the absence of cooperativity in the global picture, which demands that the expression A = \cX2KxK2 - (Kx + K2f = 4A2 - A\ = 4kx(k2 - kx) (4.10) should vanish. This condition reflects an important property of the partition function. Let us rewrite W in terms of the coefficients a as follows (4.11) W(x) = (1 + alX)(l + oc2x) The coefficients can be real and positive, or complex conjugate. The separation between these two cases is found by equating terms between (4.11) and (4.1b), i.e. ofi +
(4.12b)
The as are therefore obtained as the solution of the quadratic expression a2 - (Kx + K2)a + c12KxK2 = 0
(4.13)
as ah2 =
2
-
=-!_-
(4.14)
The term A in eq (4.14) is the same as A in eq (4.10). Hence, the condition giving rise to positive cooperativity in the global description (A > 0) coincides with the condition for the as to be complex conjugate (Briggs, 1983, 1984, 1985). If the coefficients a of the partition function are real and positive, the system cannot be positively cooperative. This conclusion is a consequence of arguments discussed in Section 2.4 about the variance of the B/N distribution. It should also be noted that the as provide information on the sum Kx + K2 = Ax and the product
4.1 The case N = 2
161
Ci2KiK2 = A2 only. As expected, these coefficients are sufficient to resolve global, but not site-specific parameters. The as written in terms of polar coordinates assume a particularly interesting form. When they are complex conjugate, then A > 0 and eq (4.11) can be rewritten as follows xp(x) = [1 + rexp(i
(4.15)
Hence, r = \lc12KxK2 = V~A^ 1
w = arccos — .
(4.16a) = arccos —-^—
2\lKK
(4.16b)
2\fM
The angle cp is bounded from 77/2, obtained for c12 —> °°, to zero, corresponding to A = 0. In the case of negative macroscopic cooperativity, A < 0 and the two coefficients a are real and positive and defined by eq (4.14). In this case, the as correspond to the norm of their polar coordinates in the complex plane since the angles vanish. When the as are real, the partition function factors out in terms of two binomials with positive coefficients, ax and a2. This is also the case when c12 = 1. Hence, analysis of the global properties cannot distinguish the case of two independent sites that bind with different affinities, Kx and K2, from the case where the two coefficients, ax and a2, are real and positive, with A < 0 . As discussed in Section 3.1, A < 0 does not necessarily demand c12 = 1. For p < 1, any value of c12 < (1 + p)2/4p yields A < 0. The case of negative macroscopic cooperativity is therefore most ambiguous and misleading when it comes to unraveling the pattern of microscopic cooperativity even in qualitative terms. If the partition function factors out as in eq (4.11), with oc\ and a2 real, interactions between the sites cannot be distinguished by a situation where the two sites are independent and bind the ligand with affinities given by eq (4.14). Before we analyze the site-specific properties of the system, it is important to discuss the global quantities X, B and XK in some detail. These quantities are given by =
din* B=
dX
dlnx
=
AlX
A xx
S
1 + Axx + A2x2 + 4Alx
+ A A x
i*
(1 + Axx + A2x2f
(4 (4.19)
2 + A XX
4 Specific cases and applications
162
and are depicted in Figures 4.1-4.3 for the case of Ca2+ binding to calbindin (Linse et ai, 1988, 1991; Martin et ai, 1990). The mean ligand activity of the macromolecule is xm = l/yA^. This is also the value of x that yields X = 1, or the half saturation of the macromolecule. We recall that the curve X is always symmetric for N = 2. The binding capacity at half saturation reaches an extremum value of 1
1
(4.20)
The Hill coefficient derived from eq (2.45) is given by _
_ i
2A X
•
A x
2
i
2A2x
_ ,
2 + A xx
,
= 1+
k xx
k 2x
(4.21)
k2x
q
IT)
d
q 6 -10.0
-9.0
-8.0
-7.0
-6.0
2+
log [Ca ] (M) 2+
Figure 4.1 Ca binding curve of calbindin (continuous line) and its mutant E17Q (dashed line). Curves were drawn according to eq (4.17) with parameter values Ax = 2.0 x 108 M" 1 and A2 = 7.9 x 1016 M" 2 for the wild-type, and Ax = 2.5 x 107 M" 1 and A2 = 3.2 x 1015 M" 2 for the mutant (Linse et ai, 1988, 1991).
4.1 The case N = 2
163
00
d
CO
d
CN
d
q d -10.0
-9.0
-8.0
-7.0
-6.0
log[Ca 2 + ] (M) Figure 4.2 Ca2+ binding capacity of calbindin (continuous line) and its mutant E17Q (dashed line). Curves were drawn according to eq (4.18) with parameter values as in Figure 4.1.
As expected, the Hill coefficient tends to one for x - » 0 and x-+ oo ? is positive everywhere and is bounded from zero to two. The extremum value of nH reached at half saturation (4.22)
1 + V*l/*2 confirms that the asymptotic values of nXj2 are zero and two, reached for extreme negative and positive macroscopic cooperativity respectively. The variance of the B/N distribution is also noteworthy. If ocx and a2 are real, the term o\ in eq (2.97) vanishes and the variance is
-
1 , ^ + V^A"
(4.23) -In2 4 Ax-yj-t where we recall that o\ — TT2/3 and A < 0. If ocx and a2 are complex 2
164
4 Specific cases and applications o d
q
q co
q CD
-10.0
-9.0
-8.0
-7.0
-6.0
log [Ca 2+ ] (M) Figure 4.3 Affinity plot for Ca2+ binding to calbindin (continuous line) and its mutant E17Q (dashed line). Curves were drawn according to eq (4.19) with parameter values as in Figure 4.1. conjugate, the term o2r in eq (2.97) vanishes, the angle cp is given by (4.16b) and the variance is = On — arccos z
(4.24)
The value of a2 is always greater than a0 if A < 0 and can grow without bounds. In the case A > 0, the variance is always less than a\ and reaches the minimum value of TT2/12 when cooperativity is a maximum. The global parameters Ax and A2, or kx and k2, can be derived from analysis of X, B or XK. Resolution of the site-specific parameters Ku K2 and en, on the other hand, demands measurements of at least one site-specific property of the system. The problem of resolving these parameters can be understood using contracted partition functions. Since any response function X, B or XK is derived by differentiation or transformation of V, all the information for the resolution of binding
4.1 The case N = 2
165
Table 4.1. Contracted partition functions and weighting factors for N = 2. Order 0 1 1 1 1
Contracted partition function
Weighting factor
W(x) = 1 + ( # i + K2)x + c12K1K2x2 oii^*) = i + K2X 0x V2(x) = l + JSTuc 1 WI(JC) = 1 + c12K2x ^ ( x ) = 1 + cnKxX
1 1 1 K xx K2x
parameters is stored in the partition function. The function W can provide two such parameters, since it is a polynomial of second order. If three parameters are to be resolved, an additional constraint is needed. This constraint is provided by a contracted partition function. The possible contracted partition functions for N = 2, up to order TV - 1, and their proper weighting terms are listed in Table 4.1. We recall from Section 3.2 that the TVth-order contracted partition functions are all equal to one and are weighted by the 2N terms defining the partition function W. The Nth-order contracted partition functions are therefore omitted from the table. The table shows that W can be written in a series of alternative forms as follows xp(x) = ^ ( J C ) + ^ ( J C ) * ! * = °V2(JC) + l^2{x)K2x (4.25) Each first-order contracted partition function can itself be expressed in terms of second-order contracted partition functions as follows o ^ ( x ) = MW12(x) + 0lW12(x)K2x = 1 + K2x (4.26a) °W2(x) = °°Wi2(x) +1(^Un(x)K1x = 1 + Kxx (4.26b) n (4.26c) (*) + W12(x)c12K2x = 1 + c12K2x n (x) + W12(x)c12KlX = 1 + c12KlX (4.26d) Knowledge of any of the first-order contracted partition functions in Table 4.1, in connection with W, is sufficient to resolve the three independent site-specific parameters Ku K2 and ci2. This can be done by measuring Xx or X2 given by the expressions Xl
= KLx
X2 =
W(x)
W(x)
l + (Kt + K2)x -
(4.27a)
K2)x + cl2KxK2x2 (4.27b)
166
4 Specific cases and applications
with Xx + X2 = X. These functions are illustrated in Figure 4.4 for Ca2+ binding to the two sites of calbindin. The numerator of Xx contains a term which is a function of Kx only, and the same holds for X2 relative to the term K2. In the limit x -* 0, Xx grows linearly as Kxx and the site-specific parameter Kx can be determined in a direct way. Once Kx is known, K2 and cX2 can be decoupled from knowledge of Ax and A2. We note from the definitions (4.27a) and (4.27b) that Xx and X2 can be derived from the partition function W as implied by eq (3.50), i.e., din K; where 7 = 1,2. The coefficients vf/- = 3 In at/d In definition (4.14) and form the matrix
(4.28) are derived from the
00
d
cq d
CN
d
q d -10.0
-9.5
-9.0
-8.5
-8.0
-7.5
-7.0
log [Ca 2+ ] (M) Figure 4.4. Site-specific Ca2+ binding curves of site 1 (continuous line) and site 2 (dashed line) of calbindin. Curves were drawn using parameter values obtained from the analysis of structural perturbations, as discussed in Section 4.5. The site-specific parameters are: Kx = 1.7 x 108 M"1, K2 = 2.7 x 107 M"1 and c12 = 18 (see also Table 4.16 and Figure 4.22).
4.1 The case N = 2 Kx —
K
:x + K2 + V^A
V =
:t +
K2
2
—
A2
KX + K2 + y^A
\
(4.29)
K,
-A \
167
- V-A
- V -i
with A{ = 2cX2KxK2 - KX{KX + K2) (4.30a) A^ = 2cnKxK2 - K2{KX + K2) (4.30b) It is easy to verify from the relationships above that the matrix of the coefficients vhas columns and rows whose elements add up to one. The definition of the coefficients v yields the analytical expressions for the first two moments of the Bx and B2 distributions. The relevant expressions for the first moment are = lnx m - I - - v21lln — a22 \2 / K2-
(\nx2)
=\nxm
.
Kx
z^ i n •
Kx + K2 + V - A
- { - - v22 I n \2 / oc2
In
2V-A
(4.31a)
Kr - V - A K2
-A x + K2 - V - A
(4.31b)
for oci and a2 real and positive (A < 0). These are the expressions for the mean ligand activity of the site-specific binding curves Xx and X2. For <xt and a2 complex conjugate (A > 0), one has the somewhat simpler relationships (InXi) = I n x m — I - — v 211 In — - = I n x
\2
J
arccos
a2
= lnx m - I - - v 2 2 lln— = \2 / a2 In either case, the values of
^ i -
2\lcx2KxK2 (4.32a)
arccos (4.32b) and ln;cm2 = (lnx 2 ) are
168
4 Specific cases and applications
symmetrically disposed around the mean ligand activity of the macromolecule as a whole, since lnx ml + \nxm2 = 21nxm
(4.33)
The variance of the Bx and B2 distributions can be found from application of (3.149) and is given by oc2
3
for arx and a2 real and positive (A < 0). For ocx and <*2 complex conjugate (A > 0) one has o\
= o\ - c? = £ - arccos2 3 =
^ ^ 3
arccos2
Kl
+ Kl
(4.35a)
2^KK ^
+
^2 2^KK
(4.35b)
The interesting result is that ox = o2 for N = 2, regardless of the sign of A. The binding capacities of the individual sites always have the same variance, regardless of the cooperativity pattern. In the case of macroscopic positive cooperativity, the variance of the Bx and B2 distributions also coincides with the variance of the B/N distribution for the macromolecule as a whole, given by eq (4.24). This shows quite simply the nonadditive character of the variances, and hence cooperativity, in the site-specific scenario. For example, in the case c12 = 1, the site-specific variances are equal to 7T2/3, but the variance of the B/N distribution can grow without bounds with the heterogeneity of the sites. For practical purposes the site-specific parameters Ku K2 and c12 can be resolved from analysis of Xx or X2 using the properties of the site-specific affinity functions
K, = •K, =
; ^ l = *,1 + C ^ * »W,U) 1 + K 2x
(4.36a)
*» = K 2^M - K,1 + C"Kl" (1 - X2)x
(4.36b)
*' = (1 - X{)x
K
4.2 The case N = 3
169
The values of Kx and c12 can be determined from XKX in the limits x —> 0 and x —> oo, since °K1 = K1 and ^K1 = c12Kx. Alternatively, the values of K2 and c12 can be determined from XK2, since °K2 = K2 and °°^ 2 = ^12^2In practice, however, if K1 and K2 differ by a factor of ten or more, measurements of only one site-specific binding curve may be insufficient to resolve all three parameters. In fact, assume that K1 » K 2. If X1 is measured, the term linear in x in the denominator of (4.27a) is insensitive to K2 and only K1 and the product c12K2 can be resolved from analysis of experimental data. Since Xx saturates in a range of x values such that K2x « 1, the affinity function plot of l n * ^ versus lnx is characterized by an asymptote l n 0 ^ = In Kx which allows for the determination of Kl9 but no asymptote for x large. This prevents measurements of c12 and hence K2 separately. In the case where Kx » K 2 it is X2, and not Xu that should be measured experimentally. Although the denominator of (4.27b) may still be insensitive to K2, the numerator grows linearly in K2. Also, the affinity function plot yields the value of K2 directly from the asymptote for x —> 0. The value of K1 is derived from the term linear in x in (4.26b). Once K1 and K2 are known, c12 is derived from the term quadratic in x in (4.27b) without difficulty.
4.2 The case N = 3 The case N = 2 has the unique advantage of being mathematically tractable. However, it by no means represents a paradigm for the complexity arising from cooperative interactions in the global and local descriptions. In fact, the global binding isotherm for N = 2 is always symmetric, while the site-specific isotherms are always asymmetric. Hence, the effect of interactions and cooperativity on the shape of the response functions X cannot be studied for N = 2. Another important feature of site-specific binding processes, regarding the nonapplicability of thermodynamic stability to the properties of the binding capacity, does not apply to the case N = 2 either. The simplest cooperative system that encapsulates paradigmatically all the important aspects of global and site-specific binding processes is provided by a macromolecule containing three sites. The partition function for the case N = 3 can be written in the global and local descriptions as W(x) = 1 + Axx + A2x2 + A3x3 W(x) = 1 + (K, + K2 + K 3 )* + ( c 1 2 ^ ^ 2 + c13K,K3 + + cl23KxK2K3x3
(4.37a) c23K2K3)x2 (4.37b)
170
4 Specific cases and applications
Again, from the uniqueness of W it follows that Ai = Kx + K2 + # 3 (4.38a) A2 = cnKrK2 + c13KtK3 + c23K2K3 (4.38b) A 3 = c123K1K2K3 (4.38c) There are a total of eight distinct configurations in the local description, corresponding to the eight terms of the partition function (4.37b). These configurations can be associated with the vectors [000] [100] [010] [001]
[no] [ioi] [on]
(
[111] As usual, the unligated form of the macromolecule, M, is the same as the configuration MQOO labeled in a site-specific fashion and is used as reference in the definition of the partition functions (4.37a) and (4.37b). The singly-ligated form of the macromolecule in the global description, MX, is the sum of the site-specific configurations M100, MOio and Mooi, corresponding to the intermediates with site 1, site 2 or site 3 ligated. The doubly-ligated form, MX2, is the sum of M110, M1Oi and MOn, while the triply- or fully-ligated form, MX3, is the same as M m . Since there are eight configurations in eq (4.37b) and one is used as reference, then there must be seven independent reactions among the possible intermediates leading to a total of seven independent equilibrium constants. These are three site-specific binding affinities Ku K2 and K3, three interaction constants c12, c13 and c23 for second-order coupling and one constant c123 for third-order coupling. These constants are associated with the reactions (4.40a) (4.40b) MoooH Moio (4.40c) MoooH-Xo Mooi (4.40d) M 1 0 0 + IV*010 ^ Mooo + Mno (4.40e) M 1 0 0 + h -••001 ^ " ^ Mooo + Mioi (4.40f) M 010 + Ik^ • 0 0 1 ^ ^ MQOO + Mon (4.40g) M 100 + Moio + ^^001 O ZJVlooo < J^»-iii In the first reaction site 1 is ligated, while sites 2 and 3 are kept unligated, and the equilibrium constant is Kx. Likewise, in the second reaction site 2 is ligated, while sites 1 and 3 are kept unligated, anc1 the equilibrium MQOO H
Mioo
4.2 ThecaseN = 3
111
constant is K2. In the third reaction site 3 is ligated, while sites 1 and 2 are kept unligated, and the equilibrium constant is K3. In the fourth reaction a dismutation is considered whereby the unligated and doubly-ligated intermediate with sites 1 and 2 ligated are generated from the two parent singly-ligated configurations. The equilibrium constant for this reaction is c12 and is a measure of the linkage between sites 1 and 2. In the fifth and sixth reactions analogous dismutations are considered for the site pairs 1 and 3, and 2 and 3, with equilibrium constants c13 and c23. The seventh and last reaction is also a dismutation, but it involves the generation of the reference unligated form and the fuUy-ligated intermediate from the three parent singly-ligated species, and the equilibrium constant is the thirdorder interaction constant c123. The global parameters Au A2 and A3 refer to the reactions M + X o MX (4.41a) M + 2X o MX2 (4.41b) M + 3X o MX3 (4.41c) The three stepwise binding constants ku k2 and k3 refer to the reactions M + X ^ MX (4.42a) (4.42b) MX + X o MX2 MX2 + X o MX3 (4.42c) with Ax = 3kx (4.43a) A2 = 3kxk2 (4.43b) A3 = kxk2k3 (4.43c) and conversely kx = Ax/3 (4.44a) k2 = A2IAX (4.44b) k3 = 3A3/A2 (4.44c) In the absence of interactions among the sites, all site-specific interaction constants are equal to one and the partition function simplifies into the factorial form W(JC) = (1 + Kxx){\ + K2x){\ + K3x) (4.45) In general, the partition function can be written as follows W(JC) = (1 + axx)(l + a2x)(l + a3x) (4.46) where the as can be all real and positive, or one real and positive and two
172
4 Specific cases and applications
complex conjugate. The separation between these two cases is found by equating terms between (4.46) and (4.37b), i.e., (4.47a) ax + a2 + a3 = Kx + K2 + K3 axa2 + axa3 + a2a3 = c12KiK2 + cX3KxK3 + c23K2K3 (4.47b) axa2a3 — c123K1K2K3 (4.47c) In general, the as are very complicated expressions of the site-specific parameters. The global quantities X, B and XK are dlnW = Axx + 2A2x2 + 3A3x3 dlnx 1 + Axx + A2x2 + A3x3 Axx + 4A2x2 + {AXA2 + 9A3)x3 x
5
dX dlnx
=
(1 + Axx + A 2 x 2 +
A2x2
A3x3)2
(4.49) (4.50)
and are illustrated in Figures 4.5-4.7 for the case of the A-repressor binding to its operator sites O R 1, O R 2 and O R 3 (Ptashne, 1986; Senear and Ackers, 1990). The mean ligand activity of the macromolecule is xm = A 3 and, in general, does not coincide with the value of x at half saturation. This is seen only when X is symmetric, or else when A3 = {A2/A{)3, or equivalently kxk3 = k\, as demonstrated in Section 2.3. The patterns of macroscopic cooperativity can be understood from the properties of the affinity function. It is convenient to rewrite XK in terms of the stepwise binding constants as follows
•K = t l l + *** + W
(4.5!)
1 + 2kxx + kxk2x2 The asymptotic values of XK are °K = kx and °°K = k3. The ratio k3/kx defines positive (k3/kx > 1) or negative (k3/kx < 1) macroscopic cooperativity. The case kx = k2 = k3 corresponds to the absence of macroscopic cooperativity. Unlike the case N = 2, the conditions for macroscopic cooperativity do not provide unambiguous answers to the cooperativity pattern of the global description. Macroscopic positive cooperativity implies k3 > kx, regardless of the value of k2. The macroscopic pattern of cooperativity remains unchanged when k2 ^ ku or kx< k2< k3, or else k2 ^ k3, although different values of k2 bring about significant differences in the properties of the macromolecule. Comparison of consecutive
4.2 ThecaseN = 3
173
q CO
q
LO
d
q d -15.0
-13.0
-11.0
-9.0
-7.0
-5.0
log [Repressor Dimer] (M) Figure 4.5 Binding curves of the A cl repressor to its operator at pH 5.0 (continuous line) and pH 8.0 (dashed line). Curves were drawn according to eq (4.48), with parameter values Ax = 3.9 x 1012 M"1, A2 = 2.1 x 10*5 M"2, A3 = 2.7 x 1035 M"3 for pH 5.0, and Ax = 1.95 x 109 M"1, A2 = 9.7 x 1018 M"2, A3 = 4.8 x 10^5 M"3 for pH 8.0 (Senear and Ackers, 1990).
stepwise constants reveals the stepwise pattern of cooperativity. When k3> k2> ku the binding affinity increases uniformly with ligation. On the other hand, when k3 > kx > k2 the binding affinity first decreases and then increases. Although the overall result is that the third ligation step takes place with higher affinity than the first, the second step occurs with a lower affinity than the first. When k2 > k3 > kx, the binding affinity first increases and then decreases. Again, the overall result is that the third ligation step takes place with higher affinity than the first, but the second step now occurs with an affinity larger than the third. In the first case, the stepwise pattern of cooperativity is uniform. In the other two cases, we speak of mixed cooperativity in the global description. The possible
174
4 Specific cases and applications
00
d
CO
d
q d -15.0
-13.0
-11.0
-9.0
-7.0
-5.0
log [Repressor Dimer] (M) Figure 4.6 Binding capacity of the A cl repressor operator as a function of the logarithm of repressor concentration. Curves were drawn according to eq (4.49) with parameter values as in Figure 4.5.
patterns of cooperativity for N = 3 are listed in Table 4.2. A 4- sign indicates the presence of positive cooperativity in the stepwise sense, a — sign refers to negative cooperativity and 0 indicates the absence of cooperativity. Interestingly, the same pattern of stepwise cooperativity can give rise to two different macroscopic patterns. The case of A cl repressor binding to its operator shown in Figure 4.7 is a clear example of mixed cooperativity, with the affinity first increasing and then decreasing to give a macroscopic pattern of negative cooperativity. When there are three stepwise binding constants, the sign of stepwise cooperativity can change only once (Whitehead, 1980b, 1981). This is confirmed by the properties of the affinity function. A change in the stepwise pattern implies the existence of an extremum in the affinity plot.
4.2 ThecaseN = 3
175
Table 4.2. Patterns of stepwise and macroscopic cooperativity for N = 3. Pattern *1
=
k2 = kz
k2> k3 ki > k3> k2 k2 h> kl< kk2 ki< 2 k\< k3
Cooperativity
Stepwise
Macroscopic
Uniform Uniform Mixed Mixed Mixed Mixed Uniform
00
None Negative Negative Positive Positive Negative Positive
q
CO
S o *
o q oo
q -15.0
-13.0
-11.0
-9.0
-7.0
-5.0
log [Repressor Dimer] (M) Figure 4.7 Affinity plot for A cl repressor binding to its operator. Curves were drawn according to eq (4.50) with parameter values as in Figure 4.5. Note the macroscopic pattern of negative cooperativity, with mixed (+—) stepwise cooperativity due to k3 < k\ < k2 (see also Table 4.2).
176
4 Specific cases and applications
The slope in the affinity plot, AX = d l n ^ / d l n x (see Section 2.2) is 2k2x(l
+ k3x)
2M(1 + k2x) 2
1 + 2kxx + fci*2*2 1 + 2k2x + k2k3x The expression giving the extrema of \wxK is obtained for AX = 0 and is given by kxk2{k3 - k2)x2 + k2(k3 - kx)x + k2 - kx = 0 (4.53) Never does this equation have two positive real roots, indicating that the slope of the affinity function can change sign only once for N = 3. The Hill coefficient derived from eq (2.44) is given by 2k2x(l H
+ k3x)
1 + 2k2x + fc2A:3x2 1 + 2kxx +
k t k 2x 2 (4.54) The Hill coefficient tends to one for x^>0 and x ^ o ° , is positive everywhere and is bounded from zero to three. When the binding curve is symmetric, nH has an extremum value at x = xm given by fcl n1/2 = 1 + 2k2 + k* - 4 (4.55) k2 + 3iki 3k2 + k3 The upper bound of n H is reached in the limit k3 » k 2 » * x and is equal to three. Resolution of the site-specific parameters Ku K2, K3, c12, c13, c23 and c123 demands measurements of at least two site-specific properties of the system. This is because the global partition function provides one constraint for these parameters through eqs (4.38a)-(4.38c), but Ax and A2 contain three independent terms, thereby demanding two additional constraints. These constraints are again provided by contracted partition functions. The possible contracted partition functions for N = 3, up to order N — 1, and their proper weighting terms are listed in Table 4.3. The partition function W can be written in a series of alternative forms as follows
(4.56) In addition, each contracted partition function up to second order can
4.2 The case N = 3
111
Table 4.3. Contracted partition functions and weighting factors for N = 3. Order O
Weighting factor
Contracted partition function \TJ/ V .\
1 j _ /IT'
* \X ) —
I
J7"
_L
f
\
v
J. Z'/'
V
V
-1- s*
V
V
J- ' v - ^ 1 ' -**-2 ' *^3)X ' V^12-**-l-**-2 ' C^3xVi/V3
1
1
"+" C23K2K2JX + C123K1K2K3X
^ ( x ) = 1 + (K2 + K3)x + c23K2K3x2
1
K J
1
t 2\X)
1
1
ll
1 1 1
=
1 "I" v ^ l ' ^ 3 ^ ' ^13^1^.3-^
0\Tf / v \ v < /
1 _ L / ' t ^ _ L ^ ' ^ v _ L ^
t'3( X ; — 1 T" \J±i ~r r±2)X
f
^
V2
' C12A1A2X
1 J1
1
P l ( x ) = 1 + (C12K2 + Ci3iC3)x + Ci23^2^3^ 2 ^ 2 ( ^ ) = 1 + (Ci2^l "I" CyzK^jX -h C123K1K2X M*3(x) = 1 + (Ci3^i + C23K2JX + C123K1K2X
K\X K2X K3X
°°Vu(*)= 1 + ^ 3 ^ °°Vi3(*)= 1 + ^2X °°V23(*)= 1 + KlX 01x Vn(x) = 1 + Cza^A: 01x Vn(x) = 1 + C23#2* ^ ^ ( x ) = 1 + caKix Wx ¥ u(x) = 1 + CI3^ 3 A: 10t P 13 (x) = 1 + c i 2 ^ 2 x 1Ol r I 23(x) = 1 + c12KlX " ^ ( x ) = 1 + ci23CilK3x ^13(x) = 1 + C123C13 i^2X
1 1 1 K2x ^3^ K3x /iCiX ^ix K2x
2 2 2 2 2 2 2 2 2 2 2
CI2KIK2X2
c\3K\K3x
11 f
2
C23^2^3X 2
1 23(x) = 1 + C123C23 A^lX
itself be expressed in terms of higher-order contracted partition functions. For example, 01
W12(x)K2x
= ™V
13(x)
+ 01Wn(x)K3x
(4.57b)
w
(4.57c)
(x) + W23(x)Kix (x) + W23(x)K2x =
000
°°W 13(x) =
000
m
mx
(4.57a)
01
W123(x) +
001
W123(x)^3X = 1 + ^ 3 x
(4.57d)
W123(x) +
010
W123(x)*:2x = 1 + K2x
(4.57e)
m
W23(x) = ¥ 123(x) + ^n3(x)KlX = 1 + Kix (4.57f) Knowledge of any two of the first-order contracted partition functions in Table 4.3, along with W, is necessary and sufficient to resolve the seven independent site-specific parameters. These partition functions can be accessed from measurements of Xx, X2 or X3 given by the expressions Xx = Kxx[l + (c12K2 + c13K3)x + cmK2K3x2] K2 + K3)x + (cl2KxK2 + cnK.Ki + c23K2K3)x2
(4.58a)
178
4 Specific cases and applications
K2x[l + ( C l 2 *i +
C23K3)x
c23K2K3)x2
K2 + K3)x
(4.58b) K3x[l
C2iK2)x
1 + (Kx + K2 + K3)X + (012*1*2 + C13KtK3 + C23K2K3)X2 + Ci23*l*2*3* 3
(4.58c) with X1 + X2 + X3 - X. These functions are sketched in Figure 4.8 for the case of A cl repressor binding to its operator sites. Again, the
CO
d
CO
d
CN
d
-13.0
-12.0
-11.0
-10.0
-9.0
-8.0
-7.0
-6.0
-5.0
log [Repressor Dimer] (M) Figure 4.8 Site-specific binding curves for the interaction of the A cl repressor with its operator sites O R 1 (continuous line), O R 2 (dashed line) and O R 3 (chain line) at pH 7.0 (Senear and Ackers, 1990). Curves were drawn using eqs (4.58a)-(4.58c) with site-specific parameter values: ^ = 1.3 x 1010 M , K2 = 1.1 x 108 M" 1 , K3 = 1.7 x 107 NT 1 , cn = 241, c13 = 1, c23 = 801, cm = 950.
4.2 The case N = 3
179
site-specific binding curves can be derived from W by differentiation as follows
X, = ^ * «
(4.59)
where / = 1, 2, 3. The most direct way to derive the site-specific parameters is from the properties of the affinity function. Given Al9 A2 and A3 from analysis of the global properties of the macromolecule, the site-specific affinity functions
c13K3)x +
c123K2K3x2
Wi(x) X7 - X2)x
1 + (K2 + K3)x + c23K2K3x2
W2(x) X,
1 + (K1 + K3)x
(4.6ua)
" X3)x - K,31
+ (Cl3Kl
+ Cl3Kl)X
+ c
™ KlKlx2 2 1 + (Kx + K2)x + c12K1K2x
(4 60c)
yield a complete solution of the problem. The values of Kx and cU3/c23 can be determined from XKX in the limits x —> 0 and x -» oo. The values of ^ 2 and cU3/c13 can be determined from XK2. This information is sufficient to resolve all parameters since the sum Kx + K2 + K3 = Ax and the product cU3KlK2K3 = A3 can be derived from the global properties of the macromolecule. We now address the problem of thermodynamic stability at the sitespecific level recalling the results discussed in Sections 3.2 and 3.3 that the binding capacity of an individual site can assume negative values for N ^ 3. We address this problem in a purely mathematical fashion by noting that if Bj (;' = 1, 2 or 3) can be negative, then there must be a condition for which the critical value J5; = 0 is observed, as shown in Figure 4.9. When B} < 0, the binding curve X] shown in Figure 4.9(a) necessarily cuts a straight line X] = 77 parallel to the In x axis at three
4 Specific cases and applications
180
-2.0
Figure 4.9 Negative binding capacity in the site-specific binding curve for N = 3 for a hypothetical example where Ki = l, K2 = K3 = 0.1, cu = ci3 = 0.1, C12 = 100 and c123 = 100 «>), left), C123 = 10 ((a), middle) or ci23 = 1 ((a), right and (b)). Curves refer to site 1. Changing the value of cU3 affects the shape of the site-specific binding curve (a). For cm = 1, site 1 shows negative binding capacity values (b) and a straight line parallel to the lnx axis cuts the binding curve at three points (a).
4.2 The case N = 3
181
points. This is exactly what has been found experimentally in the case of electron binding to cytochrome aa3 (Kojima and Palmer, 1983), as discussed in Section 3.2 (see Figure 3.10). Hence, for B,< 0 the equation Xj• — r] = 0 has three real and positive roots. The critical value of 5 ; = 0 is observed when all roots are identical. We shall define the value of rj corresponding to this case as Xc, to denote the critical value of Xj for which Bj: = 0. Using Xx as an example, the relevant equation giving three identical and positive roots is - WOO = 0 Expanding the terms in eq (4.61) yields
(4.61)
^(JO^I*
+ (X+ - Xc)Axx
-Xc
- (Xc - X.)A2x2
+ (1 - Xc)A3x3
= 0 (4.62)
where X+ = KxjAx and X_ = K1(c12K2 + c13K3)/A2 represent the upper and lower bounds for the value of Xc compatible with the existence of three identical and positive roots in eq (4.62). Let xc be the value of x at the critical point. Then -Xc
+ (X+ - Xc)Axx
- (Xc - X_)A2x2
+ (1 -
Xc)A3x3
= A3(l - Xc)(x - x c ) 3 = 0
(4.63)
Hence, x3c = 3x2c =
^ ( l * ) A
(4.64a)
(X+
(4.64b)
"
X M l
(1
X)A
(1 -
XQ)A3
From eq (4.64a) follows the rather interesting result that the value of X c is always located on a binding curve defined as Xc =
AX C
' " 3 (4.65) 1 + A 3x c This is the binding curve of an individual site in the limit where only the unligated and fully-ligated species are significantly populated, i.e., in the fully cooperative case. Elimination of xc from eqs (4.64a)-(4.64c) yields the quadratic expression (AXA2 - 9A3)X2C - [A1A2(X+ + X.) - 9A3]XC + A1A2X.X+
=0 (4.66)
which must have only one positive solution, due to the definition of Xc.
182
4 Specific cases and applications
This requires as a necessary condition [A1A2(X+ + XJ) - 9A3]2 - AA1A2X.X+{AlA2 - 9A3) 2 = [k1(X+ + X.) - k3] - 4fc1X_X+(A:1 - k3) = 0 (4.67) which is only satisfied provided AXA2 > 9A3, or kx > k3. Quite interestingly, the binding capacity of site 1 can vanish if, and only if, the macroscopic cooperativity of the macromolecule is negative. This condition applies to any of the three sites, since it is independent of site-specific properties and is a necessary constraint to which the stepwise equilibrium constants of the system are subject. The necessary and sufficient condition for B1 = 0 is given by eq (4.67) and yields the critical value of Xx as follows x
=
A1A2(X+ + X-) - 9A3
=
k,(X+ + X.) - k3
2{AXA2 - 9A3) The corresponding value of xc is X.-
'
*.(*»
2(*x - k3)
+ *•)-*,
V Kt(2 -X+as derived from eq (4.64a).
XJ) - K3
1_
KxK2Ki
4.3 The case N = 4 This is the simplest case that shows the role of second-order contracted partition functions in the resolution of site-specific parameters. The partition function for the case iV = 4 in the global and local descriptions is W(x) = 1 + Axx + A2x2 + A3x3 + A4x* (4.70a) W(x) = 1 + (Kt + K2 + K3 + K4)x
(4.70b) Hence, Ai = Kx + K2 + K3 + K4 (4.71a) A2 = c12K1K2 + cl3KxK3 + cuKvK4 + c23K2K3 + c2AK2KA + c34K3K4 (4.71b)
A4 = d^KiK^K,
(4.71c) (4.71d)
4.3 The case N = 4
183
The total number of distinct configurations in the local description is sixteen, corresponding to an equivalent number of terms in the partition function (4.70b). These configurations can be associated with the fourdimensional vectors [0000] [1100]
[1000]
[0100]
[0010]
[0001]
[1010]
[1001]
[0110]
[0101]
[1110]
[1101]
[1011]
[0111]
[0011]
(4.72)
[1111] The unligated form of the macromolecule, M, is the same as the configuration MQOOO labeled in a site-specific fashion and is used as reference in the definition of the partition functions (4.70a) and (4.70b). The singlyligated form of the macromolecule in the global description, MX, is the sum of four site-specific configurations M1Ooo, MOioo, Mooio and MQOOI, corresponding to the intermediates with site 1, site 2, site 3 or site 4 ligated respectively. The doubly-ligated form, MX 2 , is the sum of six site-specific configurations (M 1100 , M1OiO, M 100 i, Mono* Moioi> Moon)> the triply-ligated form, MX 3 , is the sum of four configurations (M 1110, M11Oi, M1On, M o m ) , while the quadruply- or fully-ligated form, MX 4 , is the same as M i m . There are fifteen independent reactions among the possible intermediates leading to a total of fifteen independent equilibrium constants. These constants include four site-specific binding affinities Ku K2, K3 and K4 for the reactions + X o M1Ooo
(4.73a)
+ X <* Moioo
(4.73b)
+ X o MQOIO
(4.73c)
+ X^Moooi
(4.73d)
where a given site is ligated while the other sites are kept unligated. The second-order interaction constants c 12 , c 13 , c 14 , c 23 , c24 and c 34 , are associated with the dismutations MloOo + Moioo o MQOOO + M1100
(4.74a)
MloOo + MQOIO o MQOOO + M1OiO
(4.74b)
Miooo + MQOOI o Moooo + Mim
(4.74c)
+ Moolo <=> MQOOO + Mono
(4.74d)
+ Moooi <^> Moooo + MOioi
(4.74e)
+ Moooi <=> MQOOO + Moon
(4.74f)
184
4 Specific cases and applications
where the reference species and a doubly-ligated intermediate are generated from the parent singly-ligated intermediates. The third-order interaction constants c123, c124, c134 and c234, are associated with the dismutations M1Ooo + MQIOO + MQOIO o 2MOooo + M m o (4.75a) M1000 + MQIOO + MQOOI o 2MOooo + M11Oi (4.75b) M1Ooo + MQOIO + MQOOI o 2Moooo + M1On (4.75c) MQIOO + MQOIO + MQOOI o 2Moooo + Mom (4.75d) where the reference species and a triply-ligated intermediate are generated from the parent singly-ligated intermediates. Finally, the fourth-order interaction constant c1234 is associated with the dismutation M1Ooo + MQIOO + MQOIO + MQOOI O 3MOOOO + M i m (4.76) where the reference species and the fully-ligated intermediate are generated from the parent singly-ligated intermediates. The global parameters Ai, A29 A3 and A4 refer to the reactions M + X o MX (4.77a) M + 2X o MX2 (4.77b) M + 3X o MX3 (4.77c) M + 4X o MX4 (4.77d) The four stepwise binding constants ku k2, k3 and k4 apply to the reactions M + X o MX (4.78a) MX + X o MX2 (4.78b) MX2 + X o MX3 (4.78c) MX3 + X o MX4 (4.78d) with Ax = Akx (4.79a) A2=6k1k2 (4.79b) A3 =4kxk2k3 (4.79c) A4 = kxk2k3k4 (4.79d) and conversely kx = AjA (4.80a) k2 = 2A2/3Al (4.80b) k3 = 3A3/2A2 (4.80c) k4 = 4A4/A3 (4.80d)
4.3 ThecaseN=4
185
In the absence of interactions among the sites, all cs equal one and W(x) = (1 + Kxx){l + K2x)(l + K3x)(l + K4x) (4.81) The partition function written in terms of the as is W(x) = (1 + axx)(l + a2x)(l + * 3 *)(1 + a4x) (4.82) Three cases should be considered, the first where all as are real and positive, the second where two as are real and positive and two are complex conjugate, and the third where the as are two pairs of complex conjugates. In general, the expression of the as in terms of the As is very complicated and makes any algebraic manipulation hopeless. The global quantities X, B and XK are Axx + 2A2x2 + 3A3x3 + 4A4x4 =
dlnx
BD —
1 + Axxxx + A2x2 4+• A3Ax33x3 + A4x4 (AXiA 9A33)x )x33 + 4(A 4(AX,A + (A A22- +1- 9A A3 + 4A 4A44)x* )x4 Axx + 4A 4A22xx22 + A2x2 + A3x 3 + A. (A2A3 + 9]AXA4), x5 + 4A2A4x6 + A3, h Axx •+
+
xx
+ A 2x 2 H
4 + 3Axx •
A4x4) + A,x
2
A - 1 / 4 n^A
A.
coincide with the value of x at half saturation, unless X is symmetric. Patterns of macroscopic and stepwise cooperativity can be understood from the properties of the affinity function 1 + 3kxx + 3kxk2x2 + kx The asymptotic values of XK are °K = kx and ™ K = k4. The ratio k4/kx defines positive (k4/kx > 1) or negative (k4/kx < 1) macroscopic cooperativity. The case kx = k2 = k3 = k4 corresponds to the absence of cooperativity. Like the case of TV = 3, the conditions for macroscopic cooperativity provide no information on the stepwise patterns that involve the constants k2 and k3. Stepwise cooperativity can change sign twice for N = 4, as indicated by the patterns listed in Table 4.4. This is a consequence of the fact that the slope of the affinity function is 2k3x + k3k4x2)
AX = -
1 + 3k2x + 3/:2^3^2 + k2k3k4x3 3ktx(l + 2k2x + k2k3x2) 1 + 3 ^ ^ + 3kik2x2 + k1k2k3x~
(4.87)
186
4 Specific cases and applications
Table 4.4. Patterns ofstepwise and macroscopic cooperativity for N = 4. Pattern k22 = k3 = k4 k2> k3> k 4 kk2 > > kk3 < < k4 2 3 4 k2> k3< kk44 k k22> kk4 kx> k2 k4 3 k\ < k2 > k3 > k4 k\< k2> k3> k4 k\> k2< k3< k4 ki > k2 < k3 < k4 ki< k2> k3< k4 k\ < k2 > k3 < k4 ki< k2< k3> k4 k\< k2< k3> k4 kx
Cooperativity
Stepwise
Uniform Uniform Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Mixed Uniform
000
— 4— + -+-4-i
___
_ _
+-- ++ - ++ +- + +- +
4- + -
++ 4-4- +
Macroscopic None Negative Negative (k\ > Positive (ki < Negative {k\> Positive (ki < Negative (k\ > Positive (ki < Negative {ki > Positive (ki < Negative (k\ > Positive (ki < Negative {k\ > Positive (ki < Positive
k4) k4) k4) k4) k 4) k4) k4) k4) k4) k 4) k4) k 4)
and can vanish for two distinct values of JC. The Hill coefficient derived from eq (2.44) is given by 1 4- AX, or 3A:2x(l + 2k3x 4- k3k4x2) 3k2x + 3k2k3x2 + k2k3k4x3 4- 2k2x + k2k3x2) 4- 3kxk2x2 + kxk2k3x3
(4.88)
from which it follows that nn —> 1 for x —> 0 and x —> °°.The Hill coefficient is also positive everywhere and is bounded from zero to four. Resolution of the fifteen site-specific parameters demands measurements of at least five site-specific properties of the system. The global partition function provides one constraint through eqs (4.71a)-(4.71d), but A 2 contains six independent terms thereby demanding five additional constraints. Unlike the cases of N = 2 and N = 3, measurement of all site-specific binding curves Xu X2, X3 and X4 is not sufficient to resolve all site-specific parameters. This demonstrates that for TV ^ 4 site-specific binding isotherms cannot provide a complete solution of the site-specific problem. The number of independent parameters grows as C[N/2^N (Di Cera, 1994a), where [N/2] is N/2 for N even and (N + l)/2 for TV odd, while the number of constraints provided by the XjS grows as N. Clearly > C[N/2],N N for N ^ 4. Since measurements of the XjS provide informa-
4.3 The case N= 4
187
tion on contracted partition functions of first order, contracted forms of higher order must be accessed experimentally to resolve all site-specific parameters. The possible contracted partition functions for N = 4, up to order N — 1, and their proper weighting terms are listed in Table 4.5. The partition function W can be written in a series of alternative forms using first-order contracted partition functions as follows W(x) = ^ ( x ) + 1W1(x)K1x = 0xV2(x) + 1W2(x)K2x = 0xV3(x) + xW3(x)K3x = °W4(x) + xW4(x)K4x
(4.89)
Each first-order contracted partition function can in turn be written in terms of second-order contracted partition functions. For example, (x) + 01W12(x)K2x = °°W 13(x) + 01 ix) + Wu(x)K4x
01
W13(x)K3x (4.90)
For higher-order contracted partition functions one has
1 + K4x
(4.91) (4.92)
The first-order contracted partition functions in Table 4.5, along with W, can be accessed from measurements of Xu X2, X3 or X4 given by the expressions Xl
= J^-[l
+ (Cl2K2 + c13K3 + cuK4)x
+ (c123K2K3 + cmK2K4 + cmK3K4)x2
+ c1234K2K3K4x3]
(4.93a)
+ c23K3 + c24K4)x cX24KxK4 + c234K3K4)x2 + c^K.K.K.x3] W(x)
(4.93b)
+ (cnK, + c23K2 + cMK4)x + cl34KxK4 + c234K2K4)x2 + c^K^K.x3]
(4.93c)
X4 = - ^ - [ 1 + (CM*! + c24K2 + c34K3)x , + cmKxK3 + c2MK2K3)x2 + c^K.K^x3]
(4.93d)
with Xi + X2 + X3 + X4 — X. The site-specific binding curves can also be derived by differentiation as Xj — d\n W/d In Kj, with / = 1, 2, 3, 4.
188
4 Specific cases and applications
Table 4.5. Contracted partition functions and weighting factors for N = 4.
Order 0
1 1 1
W<JC) = 1 + ( * i + *2 + K3 + K4)X + (Ci2*l* 2 + Ci3*l*3 + Ci4*i* 4 + 0^*2*3 + C24*2*4 + C34K3K4)x2 + ( c i 2 3 * i * 2 * 3 + C\24K.\K2K.4 + C\34K.\K.3K.4 ' ^-234 ^ 2 ^ 3 - * ^ 4/•^' ' ^"1234-**-1 **-2**-3 "-4X — 1 + ^ A 2 T A 3 + fs.4)X + ^ C 2 3 A 2 A 3 + C24K2K4 2 + cMK3K4)x + c234K2K3K4x3 x 2 v ^ / = 1 4 \&1 4" A 3 "1" &4)X 4 (^Ci3A^iA.3 "1" C14/C1/V4 _j_ ^» t^ 1^ \ V 2 1 „ it^ t^ t^ V 3 r ~r 1 C34A.3A.4^X 0\TJ / v \ 1 / ^ _ L f _"T L lC134A-iA.3A.4X \v_L/'^» t ^ t ^ _!_•* ^ t ^ Y 3 ^ A J — I T " ^ A i T" A 2 ~r A4JA "T ^Ci2AjA2 ~r C14A1A4 i l\x)
H" C24A2A4)AC
1 1 1 1 1
1
1 1 1
~h C i 2 4 A i A 2 A 4 ^
^ 4 ( x ) = 1 -h (A^i + A^2 "1" K3JX + (C12K1K2 "1" C13K1K3
1
= 1 + ( 4 ^ 2 + C13F3 + Ci24iC34)x + (C123K2K3 + c 124K2K4 H- CI3 4 A" 3 A'4)X 2 + C1234K2K3K4X3 = ^2(-*) 1 + (Ci2A^i + C23K3 + C24A^4)x + (Ci23A^iA^3 + C124K1K4 + C234K3K4JX 4" C1234K1K3K4X 1X ¥ 3(X) = 1 4- ( C B ^ I + C23^2 + C 34 ^4)^ + (0123^1^2 3 4" C134K1K4 4* C234K2K4JX + C1234K1K2K4X ll F4(jc) = 1 4- (C14X1 + C24A^2 + C34A" 3 )A: 4 ( C I 2 4 A " I A " 2 4 Ci34A^iA^3 + C234A^2A^3)x + C1234 A^i X2 A^3X
Ktx
^(X)
^ ( x ) 00 ^13(x) °°WI 4 (JC) ™m 23(X)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
Weighting factor
Contracted partition function
= = = =
1 1 1 1
4 + + +
(tf 3 (^2 (A: 2 (^1
+ 4 + 4-
^4)^ iC4)x ^3)x A%)*
+ C34K3K4X2 4 C24K2K4X2 4 C23K2K3X2 + Ci4id^4X2
°°^24(^)= 1 4 ( ^ 4 ^ 3 ) ^ + cl3K1K3x2 I 3 4 (x) = 1 + (A"i 4 A"2)X + C^A'IA^A: 2
00l /
OllTf
/
v
\
1 _ L Z'^.
i f _ l _ ^ ^ \ - v _ L ^
^ ^ - ^ - 2
* 12\"^/ — •*• ' v^23^3 ' C24^4)X i~ C23 4 A3A 4 X ^13V^/ — 1"T ^C23A2 4- C34A4/)X 4- C23 4 A2A 4 X * 14v^/ *P23(jt) *24\^/ T3 4 (^XJ Wi2v^) 10x ¥ 13(x) 10y ¥i4(x) 1O ^F23(JC) 10 ^/ 2 4 (x)
01
10x
==
1 ~^~ \C24K2 ~^~ ^34A3)^' = 1 + ( C B ^ I + C34A"4)JC == 1 4 (^Ci4Ai "4" C34K.3JX == 1 4" (^Ci4Ai 4 C2 4 A2)^ ^ 1 4* \C\3K3 4 C I 4 A 4 ) X = 1 + (c12A:2 + c14K4)x = 1 4 (012^2 + CI 3 A"3)X = 1 4 (CnKi 4- C24A%)X = 1 4- (C12K1 4 C23A"3)A:
4" C234K.2J^3X + C134K1K4X2 4 Ci34AiA^3X 4 Ci24AiX.2-£ 4 C134K3K4X 4 cmK2K4X2 4 C123K2K3X2 4 Ci24^iJ^4JC2 + C123K1K3X2
¥ 34(x) = 1 + (ci3iCi 4 c23K2)x 4 c ^ A ^ i ^ x 2 ^P 12 (jt) = 1 4" (0123^3 4- C\24^4)Ci2X + C1234CI2 K3K4X2 ^ 1 3 ( ^ ) = 1 4 (Ci23A^2 + Ci3 4 A^ 4 )c 13 X 4 Ci234^!3 K2K4X = 1 4 (C124K2 + Ci34A^3)c14X 4 Ci234C14 K2K3X ^uix) HIT/ / V \ C^_L^. \ ^ - l v _J_ ^ ^ . - 1 ^ it^ V 2 ^ 2 3 ^ / — 1_i_/1 4 - VC123A1 4- C234t^ A4jC23 X -r Ci234C23 A1A4X == 1 4" \C\24K\ 4 C234A3)C24-^ 4" Ci234^24 A1A3AJ Y24v."^'/ * 34\-^/ = I "t" (.^134Ai 4 C234A2)C34X 4" CYI34C34 K.1K.2X " ^ m W = 1 4- tf4JC 11
K 2x K3x K4x 1 1 1 1 1 1 K2x K3x K4x K 3x K4x K4x K\X K\X K.\X K2x K2x K3x enK^x2 C13K1K3X2 C14K1K4X2 C23K2K3X2 C24K2K4X2 C34K3K4X2 1
4.3 ThecaseN = 4
189
Table 4.5. (Cont.) Order 3 3 3 3 3 J
3 3 3
m}
ra
W mW
= 1 + K3x = 1 + K2x " " f ^ O O = 1 + KiX mx ^m(x) = 1 + C34K4X mx
Vm(x)
Vm(x)
=1+
o!I iPl23(;C) = 1+ C7AK*X
Qi
ym{x) = 1 + c*K2x ^234 (x)
= 1 + C13K1X
3
3
^ S ( ^ ) = 1 + c£#2*
^m{x)
= 1 + Ci4A:4^
3
011x
^
011m /^\ _ ^ 011\T/ / v \ 1_i_y ^ " l ^ v ^234\X) — 1 1 CY34C34 A-iA 101\T/ ^v^ 1 1 . ^"^/^v
a
3 1 D 3
U (x) = 1
101I
11 l /
3
11Ol f
^
3 3
= 1 + ci^cr/A:^
PI24(JC) lOliTf / v \ 1 _i_ ^ ^-lp- v ^^134V*^'/ — -^ ' ^124^14 -*»-2'*' 110\j/ /^\= ^ .
3
3
1 1 1
CUKTX ±234\X) — I T " C14A.1X
mx
3
Weighting factor
Contracted partition function
/•
f ^ v2 C34A.3A.4A r TC TC y^
CM^I*:^2 ^ f ^ V2 C14iv 1 iv4X
°I 124(^) = 1 + x.Y£5s.
I i34(^) = 1 + C m C o 1 ^ 110\T/ / v \ -I 1 s*~]~TS' v J^234\-^/ — -»• •" ^123^-23 &-\X ^ 1 2 3 ^ j — 1 + C1234C123K4X
111\Tf / v \ I _i ^, ^ — 1 1^ v ^134V*^J — -*• ' C1234W34 A-2-* 111\Tf / v \ 1_l_y^ ^.~lt^v TT234yXJ — 1 ~r Ci234C234-/ViX
,* V Tf V V 3 C 3 4 A-i A . 3 A.4A y^ ^ ^ ^ " 1 ^ 3 ^234-^2-**-3J^4X
Resolution of all independent site-specific parameters requires two additional constraints in the form of contracted partition functions. This follows directly from the properties of the site-specific affinity functions *KX = Kxhl\xx
(4.94a)
where Xx = 1 + ( c 1 2 ^ 2 + ^13^3 + C14A:4)JC + ( c ^ A ^ + ^124^2^4 + c 134 ^ 3 A: 4 )x 2 + C 1 2 3 4 ^ 2 ^ 3 ^ 4 J C 3 and \ix = 1 4- (K2 + K3 + K4)x + 3 (c23K2K3 + c24K2K4
190
4 Specific cases and applications
(4.94b) where ^ = 1 + {c\2Ki + c23K3 + c24K4)x + (c123K1K3 + c124K1K4 + and fa = 1 + (Kx + # 3 + A:4)x + -h C \ 4 K . \ K 4 ~\~ C 3 4 K 3 J K 4 ) X
•+•
C
3
K K
3
K x
*K3 = K^llh
(4.94c)
where A3 = 1 + ( c ^ ! + c 23 ^ 2 + c34X4)a: + (0x23X^2 + cmK1K4 c27AK2K4)x
2
+ c12MK1K2K4x
3
and /% = 1 + ( ^
^
4
+
+ K2 + KA)x +
= # 4 A 4 /M 4
(4.94d)
where A4 = 1 + (c 1 4 ^! + c 24^2 + c 34 ^ 3 )x 4- (CYI^K^K^ + 0x34^^3 + 2 3 and ^4 = 1 + (A:X + A:2 + i^3)x + c23K2K3)x2 + c123K1K2K3x\ Knowledge of X i ? X2, X3 and X 4 yields the values of Ku K2, K3 and K4, determined from the site-specific affinity functions in the limit x -» 0. This information is sufficient to resolve all site-specific binding constants and the coupling constant c1234, since the value of A4 = c1234K1K2K3K4 is derived from the mean ligand activity of the overall binding curve X = Xx + X2 + X3 + X4. The ratios c1234/c234, c1234/c134, c1234/c124 and ^1234/^123 are determined from the site-specific affinity functions in the limit > 00 and yield the third-order coupling constants. However, the four x — site-specific binding curves do not provide enough constraints to resolve the six independent second-order coupling constants. Resolution of these parameters can only be obtained from contracted forms of the macromolecule where at least two sites are frozen in a particular ligation state. Two second-order contracted partition functions chosen properly, along with the four site-specific binding curves, are needed to solve the site-specific problem for N = 4. For example, ^ ^ yields c34 and ^ V ^ yields c12, thereby allowing resolution of all cs.
4.4 Site-specific properties of hemoglobin The limitation arising in connection with the case of N = 4 applies to all other cases for larger N. Structural symmetry does not eliminate this problem and hemoglobin provides a convincing and pertinent example. Human hemoglobin is composed of two identical dimers of oxygen-binding subunits, the a and /3 chains, interacting across an extended
4.4 Site-specific properties of hemoglobin interdimeric interface Lesk, Janin, Wodak together by contacts partition function for Cera, 1990, 1994a)
191
(Perutz, 1970, 1989; Baldwin and Chothia, 1979; and Chothia, 1985). Each dimer is tightly held at the ocx$\ or a2f32 intradimeric interface. The hemoglobin in the site-specific description is (Di
W(x) = 1 + 2{Ka + Kp)x + [caaK2a + CppK\ + 2(cap + c'ap)KaKp]x2 + 2(caapKa + caP/3Kp)KaKpX + CaappK aK pX
(4.95)
Here Ka and Kp denote the binding affinities of the two chains when all other chains are unligated, while the cs are the various coupling constants. A distinction between cap and c'ap should be made since the interaction of the two chains across the interface between the dimers, a(5, is to be distinguished from that within each dimer, a/3'. All models proposed for hemoglobin cooperativity (Pauling, 1935; Wyman, 1948; Monod et al.9 1965; Koshland et aL, 1966; Szabo and Karplus, 1972; Herzfeld and Stanley, 1974; Johnson, Turner and Ackers, 1984; Di Cera, Robert and Gill, 1987b; Ackers et al.9 1992) can be derived from the model-independent partition function (4.95) as special cases. 1 The connection with the overall equilibrium constants is given by Ax = 2{Ka + Kp)
(4.96a) 2
A2 = caaKl
+ cppK p + 2{cap + c'ap)KaKp
A3 = 2(caapKa + cappKp)KaKp 2
2
A 4 = caappK aK p
(4.96b) (4.96c) (4.96d)
The presence of two identical pairs of chains reduces the number of distinct configurations in the local description from sixteen to ten. Each configuration can be associated with a vector whose elements are in lexicographic order and refer to the chains in the order ax, a2, fa, jS2, i.e., [0000] [1100]
[1000]
[0010]
[1010]
[1001]
[0011]
(4.97)
[1110] [1011] [1111] f
A few years after he had left Germany, Einstein was informed that Nazi scientists had put together 100 papers to prove his theories wrong. Asked whether he had any concerns, Einstein replied: These 100 papers must be all wrong. If right, one would have been sufficient.' One can hardly think of a better comment when it comes to the endless list of models for hemoglobin cooperativity. The exact thermodynamic treatment embodied by eq (4.95) is one and sufficient.
192
4 Specific cases and applications
Symmetry of the hemoglobin tetramer imposes the following conditions [1000] = [0100] (4.98a) [0010] = [0001]
(4.98b)
[1010] = [0101]
(4.98c)
[1001] = [0110]
(4.98d)
[1110] = [1101]
(4.98e)
[1011] = [0111]
(4.98f)
The unligated form of the macromolecule, M, is the same as the configuration MQOOO labeled in a site-specific fashion and is used as reference in the definition of the partition function (4.95). The singly-ligated form of the macromolecule in the global description, MX, is the sum of two distinct site-specific configurations, MloOo and MOoio> corresponding to the intermediates with the a chain or the /3 chain ligated. In addition, there are four doubly-ligated configurations (M 1100 , M1Oio, M1Ooi, Moon), two triply-ligated configurations (M 1110 , M1On) and the fully-ligated form M i m . Since there are ten configurations in eq (4.95), one of which is used as reference, then there must be nine independent reactions among the possible intermediates and an equivalent number of independent equilibrium constants. The site-specific equilibrium constants Ka and Kp refer to the reactions M ^ o + X o MloOo (4.99a) MOOOO + X ^ M Q O I O
The coupling constants cacn cpp, associated with the dismutations
caP,
c'aP, caap,
ca^
MloOo + MQIOO OMQOOO + M1100 MQOIO
+ Moooi <*> MQOOO + MOOn
(4.99b) and caaPP are (4.100a) (4.100b)
M1000 + MQOOI <^ MQOOO + M100i
(4.100c)
M1000 + MQOIO <=> Moooo + M1OiO
(4.100d)
M1Ooo + MQIOO + MQOIO <^ 2MOOoo + M m o
(4.100e)
M1000 + Mooio + Moooi <^ 2MOOoo + M1On
(4.100f)
Miooo + MQIOO +4 Mooio + MOOOi o SMQOOO + M i m The site-specific binding curves of the two chains are X. a = Kax
(4.100g)
^ 1 + [caaKa + (caP + c'aP)Kp]x + (2caapKa +
\ (4.101a)
4.4 Site-specific properties of hemoglobin
193
(4.101b) with X = 2(X, + A »
(4.102)
as expected. Complete resolution of the nine site-specific parameters in the partition function (4.95) demands measurements of at least three site-specific properties of the system in addition to the global properties. This is because A2 contains four independent terms, thereby demanding three additional constraints. Although the intrinsic symmetry of the system has reduced the number of total configurations and independent parameters, the amount of information to be gathered from experimental measurements exceeds the limits accessible from measurements of Xa and Xp alone. These binding curves provide altogether one zero-order and two first-order contracted partition functions. An additional second-order contracted partition function is needed. The possible contracted partition functions, up to third order, and their proper weighting terms are listed in Table 4.6. As seen in the general case, the partition function W can be written in a series of alternative forms using first-order contracted partition functions as follows
W(x) = °Wa(x) + xWa{x)Kax = °Wp(x) + %(*)*>*
(4.103)
Hence, Xa
= Kax
i\j/ (x) a{
W()
}
°W
=1-
(x)
*W W()
X, - JC^r^W . 1 - ^ £ W
(4.104a)
(4.104b)
W() W() The properties of hemoglobin as a whole can be derived from those of the a or /3 chain if either system is known along with its interaction with the other chains. Each first-order contracted partition function can be written in terms of second-order contracted partition functions. If only the configurations pertaining to the a chain are considered we have (4.105a) (4.105b) (4.105c) Therefore, the properties of hemoglobin as a whole can also be derived
4 Specific cases and applications
194
Table 4.6. Contracted partition functions and weighting factors for human hemoglobin. Order
Weighting factor
Contracted partition function
W(x) = 1 + 2(Ka + Kp)x + (caaK2a + 2c'apKaKp)x2 + 2(caapKa + CaafipK aK pX
(
° ( )
2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
c'apKaKp)x
2capKaKp 3 aKpx
) (
capKaKp
+ c'apKaKp)x2 + caapK2aKpx3 1 + (c^^T^ + c ^ ^ + c'apKp)x + + cappKp)Kpx + caappKaK pX 1 + (cflsX> + c a / 3 ^ a + c'apKa)x + + 2cappKp)Kax + ca(XppK aKpx °°V
/3.(x)
= 1 + (Ka ) = 1+ + (caP (c'aPKa
^
+ Kp)x 2K«x + + c'afs)Kpx + cppKp)x
Kax Kpx
2
a pa caaKlx2
Kax fiPi
(c a / 8 £ a + c w ^ ) * + caPfiKaKpx2 + (caP + c'aP)Kax + caapKlx2 (c^^T/j + caaKa)x + caaPKaKpx2
2
( a p p _ a a a ) ) = 1 + 2caapcalaKex + (caapKa + cappKp)capX (caapKa + caPpKp)c'~plx ) = 1 + 2cappCppKax +
p p p + caappcapKaKpx2 capKaKpx2 2 + cctajpc'~pKaKpx c'apKaK.px2 2 l2
^^O) =1+ £ ""Vippix) = 1 + Kax
100x W0
mx
IW(*) = 1 + caPKpx
V « ^ ( J C ) = 1 + c'apKpx
faap(x) = 1 + cppKpx 100 ^<,w(*) = 1 + caaKax 010 V«w(*) = 1 + c^^^x " ^ ^ ( J C ) = 1 + CapKax
l
= 1+ p
) = 1 + caapc'alpKax
) = 1 + c^c^JTo* = 1+
Kpx Kpx Kax Kax
CaappCaapKpX
1 Kax Kax p
Kax Kpx caaKlx2 c'aPKaKpx2 caPKaKpx2 c'apKaKpx2 capKaKp2x2
j
CaapK aKpX 3
4.4 Site-specific properties of hemoglobin
195
from those of the two a chains, if their properties are known along with their interactions with the (5 chains. The structural symmetry of the hemoglobin molecule poses some precise limitations as to what contracted partition functions should be accessed experimentally to resolve the nine independent site-specific constants. Assume for instance that all contracted partition functions with the a chains unligated can be accessed, in addition to W. This gives a total of three partition functions, W9 °Wa and m} ¥ aa. We see from Table 4.6 that these partition functions are sufficient to resolve Ka, Kp, caapp9 caap and capp. Of the second-order interaction constants, caa and cpp can be resolved separately, but only the sum of c «l3 + c'ap can be obtained. The same result applies in the case where analogous partition functions are derived experimentally for the (3 chains. Information on the separate coupling constants cap and c'ap must be obtained from contracted partition functions where one a chain and one /? chain are frozen in particular ligation states. The standard techniques developed for measuring oxygen binding equilibria of hemoglobin (Roughton, Oris and Lyster, 1955; Imai, 1974, 1982; Dolman and Gill, 1978; Gill et al., 1987) yield global binding isotherms and binding capacity measurements. These techniques are not capable of resolving the separate contributions of the two chains, or the site-specific parameters involved in the partition function (4.95). Some qualitative information on the behavior of the two chains has been obtained by NMR measurements (Viggiano and Ho, 1979; Bondon, Petrinko, Sodano and Simonneaux, 1986; Simonneaux, Bondon, Brunei and Sodano, 1988; Bondon and Simonneaux, 1990) and by binding equilibria on hemoglobin hybrids where some of the chains carry a substitution of the Fell heme atom with Ni, Mn or Co (Ikeda-Saito and Yonetani, 1980; Inubushi, D'Ambrosio, Ikeda-Saito and Yonetani, 1986; Shibayama, Inubushi, Morimoto and Yonetani, 1987; Ackers and Smith, 1987; Spiros, LiCata, Yonetani and Ackers, 1991). The cryogenic quenching technique pioneered by Perrella (Perrella and Rossi-Bernardi, 1981; Perrella et al, 1983, 1986) has provided a real breakthrough in the experimental approach to cooperative binding by hemoglobin. Using this technique, Perrella has been able to trap some of the key intermediates of ligation with CO (Perrella et al, 1990, 1992, 1993) and to dissect the salient site-specific components of the linkage between heme ligation and proton release in hemoglobin (Perrella, Benazzi, Ripamonti and Rossi-Bernardi, 1994), also known as the Bohr effect. More recently, a detailed resolution of the CO-bound intermediates in scheme (4.97) has been obtained (Perrella, 1995). These measurements represent the first, most accurate
196
4 Specific cases and applications
Table 4.7. Distribution ofligated intermediates for CO binding to hemoglobin at 48% saturation (Perrella, 1995). Species
Fraction (%)
[0000J
iooo:: ooio
1100 1010 + [1001] 0011 1
mo 1011
[11111
41.8 ± 0.2 2.4 ± 0.1 4.9 ± 0.3 1.0 ±0.1 2.9 ± 0.2 1.4 ± 0.2 3.8 ± 0.5 4.7 ± 0.7 37 ± 2
Associated site-specific term / caaKlx2/W(x) 2(c'ap + cafi)KKpx2
y
Table 4.8. Site-specific parameters for CO binding to hemoglobin. Ka = 1.5 ± 0.1 Torr"1 Kp = 3.1 ± 0.3 Torr"1 caa = 29 ± 4 cttp + c'ap = 21 ± 4
cpp = 10 ± 2 caap = 942 ± 186 capp = 570 ± 108 caaPp = 3.1 x 105 ± 0.7 x 105
and relevant experimental determinations of the site-specific properties of the native hemoglobin molecule in its reaction with a gaseous ligand. The distribution of ligated intermediates in scheme (4.97) determined at 48% saturation with CO is given in Table 4.7. The parameters listed in Table 4.8, derived from the distribution shown in Table 4.7, make it possible to reconstruct the site-specific binding curves of the two chains. The fact that only the sum cap + c'ap could be obtained experimentally is inconsequential. It follows from eqs (4.101a) and (4.101b) and Table 4.6 that Xa and Xp depend on contracted partition functions that contain only the sum Cap + c'ap- The important response functions for the a and /? chains in their reaction with CO in the hemoglobin tetramer are shown in Figure 4.10. Notwithstanding a significant difference in binding affinity, most evident in the affinity plot (see Figure 4.10(c)), the two chains are remarkably similar in their cooperative behaviour. The site-specific Hill coefficient (see Figure 4.10(d)) is practically identical for the two chains and indicates that up to 3.4 molecules of CO are bound when either chain is ligated. The communication among the chains is characterized by a
4.4 Site-specific properties of hemoglobin
197
significant degree of asymmetry. The second-order coupling constant between the a chains is three times stronger than that between the (3 chains. Likewise, the third-order coupling constant is more pronounced when the two a chains are ligated. The stronger coupling between the a chains compensates for their reduced affinity and results in a site-specific binding curve similar to that of the /3 chain (see Figures 4.10(a), (b)). It should be noted that coupling in the hemoglobin tetramer is always positive and therefore binding to any chain always promotes binding to unligated chains in a positively cooperative fashion. The global description of CO binding to hemoglobin tells us nothing about the asymmetric interactions in the hemoglobin tetramer, or the chain heterogeneity, thereby missing a great deal of important information. To illustrate this point we compute the global binding parameters from the site-specific ones listed in Table 4.8, using eqs (4.71a)-(4.71d) and (4.80a)-(4.80d). The result is Ax = 9.2 Torr"1, A2 = 357 Torr"2, A3 = 3.0 x 104 Torr"3 and A4 = 6.7 x 106 Torr"4 for the overall constants and kx = 2.3 Torr"1, k2 = 26 Torr"1, k3 = 126 Torr"1 and k4 = 893 Torr"1 for the stepwise constants. Binding of CO to hemoglobin in the global description is characterized by uniform positive cooperativity (kx < k2< k3< k4) and symmetry {kxk4 ~ k2k3). These two features are easily accounted for by practically all models for hemoglobin cooperativity proposed to date, from the simplest ones (Pauling, 1935; Wyman, 1948) to the celebrated two-state MWC (Monod-Wyman-Changeux) model (Monod et ai, 1965). However, these models do not account for the distribution of intermediates as found by Perrella (1995) and listed in Table 4.7. As in the case discussed in Section 3.1, the important details of the local picture are swamped into the average global behavior of the macromolecule. The inadequacy of these models, including the MWC model that has been widely used in the analysis of hemoglobin cooperativity, is further demonstrated by consideration of the 'pairwise coupling' between sites within the general framework of site-specific thermodynamics. A detailed treatment of pairwise coupling is given in Section 4.6. The coupling of a given chain to the rest of the macromolecule can be analyzed in terms of the contracted partition functions listed in Table 4.6. Likewise, for any given pair of chains, oc\oc2, oc\$\, oc\$2 and $\$2, the interaction with the complementary pair can be quantified in a direct way. Here we exploit the nature of the interaction constants as related to the free energy of coupling in a given thermodynamic cycle. Consider the thermodynamic cycle in Figure 4.11. Ligation of the ocxoc2 pair is taken into consideration while the fiifc pair is kept in the unligated configuration.
198
4 Specific cases and applications
CO
d
q d -3.0
-2.5
-2.0
-1.5
0.0
-0.5
-1.0
logpCO (Torr)
oo d
q d -3.0
-2.5
-2.0
-1.5
-1.0
logpCO (Torr)
-0.5
0.0
4.4 Site-specific properties of hemoglobin
199
o
O
d -3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
logpCO (Torr)
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
logpCO (Torr) Figure 4.10 Site-specific response functions for CO binding to human hemoglobin derived from the distribution of intermediates given in Table 4.7. Continuous lines refer to the a chain and dashed lines refer to the /? chain. Experimental conditions are (Perrella, 1995): 0.1 M Hepes, 0.1 M KC1, pH 7.0 at 20 °C: (a) site-specific binding curves, (b) binding capacities, (c) affinity functions and (d) Hill coefficients.
200
4 Specific cases and applications
Ka
On nO
Ka
<=>
ft
•• nO
ft
On <=>
caaKa •• n»
Figure 4.11 Schematic representation of the hemoglobin tetramer (top) and the thermodynamic cycle illustrating the pairwise coupling between the a chains when the /? chains are kept unligated (bottom). Similar cycles can be constructed for all possible ligated configurations of the complementary pair and for all possible pairs of chains in the hemoglobin tetramer, as shown in Figure 4.12. The coupling free energy in the cycle is a model-independent measure of the communication between the a chains, given the particular configuration of the /3 chains.
The free energy of coupling (in kcal/mol) in the cycle is evidently AGC = -RTlncaa
(4.106)
This quantity reflects the linkage between the a chains when the fi chains are unligated. As there are four possible configurations for the $x$2 pair, so are there four terms as eq (4.106). Furthermore, since there are four distinct pairs of chains to be considered in the hemoglobin tetramer, so are there a total of sixteen free energies of pairwise coupling as eq (4.106) (Di Cera, 1990). These terms are summarized in Table 4.9. The relevant thermodynamic cycles are sketched in Figure 4.12. For each pair, the four possible configurations of the complementary pair are listed in the usual vector form in lexicographic order. The relationships listed in Table 4.9 make it possible to quantify the linkage between any two chains in the hemoglobin tetramer as a function of the ligation state of the other two subunits. The coupling terms are independent of the binding affinities of the two chains, Ka and Kp9 and depend solely on the interaction constants cs. They encapsulate the code for cooperative coupling among the subunits in a site-specific fashion. The importance of defining pairwise coupling free energies stems from
Table 4.9. Free energies of pairwise coupling in the hemoglobin tetramer. Pair
00
10
01
11
axa2 <XiP\ <x\$i P\Pi
-RT\ncaa - RT In crap -RT\ncaf3 -RTIn cpp
-RT In (caap/capcap) -RT\n (caap/caacap) -RT In (caap/caac'ap) -RTIn (capp/capcap)
- RT In (caap/capcap) -RT In {capp/cppcap) -RT In (capp/cppcaP) -RTIn (capp,Icapc'ap)
- RT In {^aa^/a^) -RT In (capcaapp/caapcapp) - RT In (capcaapp/caa^capp) -RTIn (caacaapp/caap)
4 Specific cases and applications
202
00
axa2
11
01
on •• BO BO
no no
OB • B BO B O
on mn
on •n •• ••
OB •• Um ••
OB • B •• • •
on #n no no
on •• n# ••
OB ••
OB • B Um • •
on ••
on •• •• ••
OB «B BO BO
OB • B
on ••
on • •
on ••
on • •
BO BO
D O no OB ••
no no on on no BO PxPi
10
on mu no no
OB OB
no BO
OB «B
no no
•• Urn OB • B
n# Um •• •D no BO • • •a no BO
•• • •
BO BO OB •• BO BO
•• • •
on on
••
OB • B •• B *
• • B# OB OB • • ••
n#
•n ••
•• ••
n#
Figure 4.12 Thermodynamic cycles for pairwise coupling in the hemoglobin tetramer. The configurations of the complementary pair are listed in lexicographic order. Symbols depict the various chains as in Figure 4.11 (top).
the possibility of deciphering the code for cooperativity in a modelindependent fashion. Experimentally derived values for pairwise coupling can be compared directly with the predictions drawn from specific models. The KNF (Koshland-Nemethy-Filmer) model (Koshland et aL, 1966) based on nearest-neighbor interactions predicts the free energies of pairwise coupling to be independent of the ligation of the complementary pair (Di Cera, 1990). The MWC model predicts the free energies of pairwise coupling to be always negative and to change with ligation of the complementary pair (Di Cera, 1990). A detailed analysis of these interaction patterns is given in Section 4.6. The partition function for the two-state MWC model is1 (Monod et al., 1965) KTx)N
+ (1 +
L + 1 f
KRx)N
(4.107)
It should be pointed out that the partition function of the MWC model was first derived by Terrell Hill in an analogous context in 1960 (Hill, 1960; see eq (7-68) at p. 143), i.e., five years before publication of the MWC paper (Monod et al., 1965). Hill's derivation was ignored in the MWC paper and in practically all subsequent treatments of allosteric models.
4.4 Site-specific properties of hemoglobin
203
The various terms in eq (4.107) represent the binding affinities of the T and R states, KT and KR, and the allosteric constant, L, that gives the ratio between the concentration of unligated molecules in the T state relative to the R state. Cooperativity arises from the fact that the macromolecule is predominantly in the low affinity T state at low saturation and progressively switches to the high affinity R state as ligation proceeds. Both states are assumed to be reference systems. The T—»R transition involves all sites in a concerted fashion and brings about cooperative behavior, as in the example dealt with in Section 2.6 for the linkage between two ligands (see eq (2.162)). The free energies of pairwise coupling for the MWC model are derived from (4.107) as (Di Cera, 1990) (4.108a)
(Lc + I) 2 AG10 = AG01 = - i ? r i n ( L c AG n = - ^ ( ^
+ 1 ) ( L c + 1) 2
(Lc + I)
2
(4.108b)
+ 1 X ^ + 1) (Lc3 + I) 2
where c = KT/KR < 1 by definition. The various free energy terms in eqs (4.108a)-(4.108c) give the pairwise coupling between any two chains when the other two chains are unligated, singly-ligated or doubly-ligated. The free energy terms are always negative because (4.109) (Ld + l)(Ld~2 + 1) > (Ld~l + I)2 for any value of /, as long as c > 0. Furthermore, pairwise coupling is large at the switchover point, where Lcf ~ 1, and tends to zero for values of j ¥= s.
The global CO binding curve constructed from Perrella's data (Perrella, 1995) can be represented very well in terms of the MWC model with c « 0.0026 and L « 9 . 5 x 10 4. This gives a switchover point of s « 2, i.e., the T —» R transition takes place after two CO molecules are bound to the T state, as expected for a symmetric binding curve in the case N = 4. The free energies of pairwise coupling derived from these parameter values are: AGQO = -0.5 kcal/mol, AGi0 = AGOi = -2.6 kcal/mol, AG n = -0.3 kcal/mol. As expected, the coupling between any two chains is maximum when the T —> R transition is bridged in going from the singly-ligated to the triply-ligated species. The values for the pairwise coupling derived from Perrella's data are listed in Table 4.10, assuming that c «p = c'ap = 10.5, or cap = 1 and c'ap = 20 as extreme cases. The free
204
4 Specific cases and applications
Table 4.10. Free energies ofpairwise coupling (in kcal/mol) for CO binding to hemoglobin.
Pair CaB = OC\OCl 0Cl{5\
00 C'ap = 10.5
0C\$2 CaB = OC\OCl
P1P2
1, c'ap = 20
-2.0 -1.4 -1.4 -1.3
10
01
11
± 0.2 ±0.2 ±0.2 ±0.2
-1.2 -0.7 -0.7 -1.0
±0.2 ± 0.2 ± 0.2 ±0.2
-1.2 -1.0 -1.0 -1.0
±0.2 ±0.2 ±0.2 ±0.2
-1.3 -1.0 -1.0 -1.3
±0.2 ±0.2 ±0.2 ±0.2
-2.0 ± 0.2 -1.7 ±0.2 0.0 ± 0.2 -1.3 ±0.2
-2.2 -2.0 -0.3 -1.9
± 0.2 ± 0.2 ± 0.2 ±0.2
-2.2 -2.3 -0.6 -1.9
± 0.2 ± 0.2 ± 0.2 ±0.2
-1.3 ±0.2 -1.4 ±0.2 0.3 ± 0.2 -1.3 ±0.2
energies of pairwise coupling are practically all negative, indicating positive coupling between pairs at all stages of ligation. However, in no case is the pattern predicted by the MWC model observed. In the case Cap = cap = 10.5, the pattern is almost independent of the ligation of the complementary pair and suggests the presence of direct interactions of the nearest-neighbor type. In the case cap = 1 and c'ap - 20, coupling tends to increase slightly at intermediate saturation and remains strong at low and high saturation for all pairs but oc\$2, contrary to the predictions of the MWC model. Although a conclusive answer as to the precise code for cooperative CO binding to hemoglobin must await the resolution of cap and c'ap separately, it is clear from the data in Table 4.10 that the MWC model cannot account, even qualitatively, for the precise mechanism of hemoglobin cooperativity. There is no evidence from Perrella's data of the existence of a concerted transition between T and R. The transition probably takes place gradually with saturation, with the chains communicating directly through nearest-neighbor interactions, as predicted by the KNF model. The conclusion drawn for CO binding provides important insights into the mechanism of hemoglobin cooperativity and is anticipated by the results obtained by Ackers on the distribution of ligation intermediates using model systems (Ackers et ah, 1992). Construction of these systems has been made possible by the fundamental discovery due to Bucci and Fronticelli (1965) on the separation of the hemoglobin chains. In these systems, the ligated or unligated states of the heme Fell are mimicked by ad hoc substitutions. A remarkable finding on these systems is that the intermediates of ligation distribute themselves according to a 'symmetry
4.4 Site-specific properties of hemoglobin
205
rule' for cooperative switching. The rule is cast within an MWC-like framework and states that any intermediate can exist in either one of two quaternary states, T (low affinity) or R (high affinity). Switching from T to R is combinatorial and intermediates with ligated chains across the interdimeric a/3 interface are in the R state, while those with ligated chains on the same side of the interdimeric a/3 interface are in the T state (Daugherty et ai, 1991; Ackers et al, 1992; Doyle and Ackers, 1992; Ackers and Hazzard, 1993; LiCata, Dalessio and Ackers, 1993). The elegance of this rule stems from its simplicity. Since ligation must eventually involve chains across the interdimeric afi interface, the T—>R transition becomes an 'inevitable' consequence of the rule and so is cooperativity. The ensemble of intermediates partitioned according to the symmetry rule is shown in Figure 4.13. Resolution of the free energy levels relative to all intermediates allows for the calculation of all the free energies of pairwise coupling. The code for cooperativity is then 'broken' by inspection of the pairwise coupling pattern, independent of any mechanism. The results are given in Table 4.11 for three different model systems. In the Fell/Felll-CN system, the bound state of the heme Fell is mimicked by a cyanide bound to the oxidized Felll (Smith and Ackers, 1985). In the Fell/Mnlll system, Mnlll mimicks the ligated heme Fell (Smith, Gingrich, Hoffman and Ackers, 1987). In the system Coll/FellCO, the unligated Fell is mimicked by Coll (Spiros et al, 1991). Despite
on no ••
on
no
••
BO
BO
T state R state
•D ••
•• DO
••
••
OH BO
••
BO
• • • • Figure 4.13 Partitioning of the ten ligated intermediates of hemoglobin according to the symmetry rule (Ackers et ai, 1992). Ligation within the oc\$\ or a2fi2 dimers does not affect the quaternary state. On the other hand, ligation across the a/3 interdimeric interface induces the T-» R transition, regardless of the ligation state. Symbols depict the various chains as in Figure 4.11 (top).
206
4 Specific cases and applications
Table 4.11. Free energies of pairwise coupling (in kcal/mol) for hemoglobin model systems. Pair
00
10
Fell/Felll-CN (Huang and Ackers , 1995) OCx 0C2 0.0 ± 0.2 -0.3 ± 0.2 a i Pi -3.2 ± 0.2 -2.9 ± 0.2 0C\ij2 0.2 ± 0.2 -0.1 ± 0.2 tj\p2 -0.2 ± 0.2 0.0 ± 0.2 Fell/Mnlll (Ackers, 1990) (X\(X2 1.0 ± 0.5 0.2 ± 0.6 -3.2 ± 0.5 -4.0 ± 0.6 OC\&\ 0.0 ± 0.5 -0.8 ± 0.6 OC\$2 -1.2 ±0.5 -0.6 ± 0.6 CoIl/Fell-CO (Spiros et al., 1991) OC\OC2 0.1 ± 0.4 0.0 ± 0.6 OC\$\ -1.4 ±0.5 -1.5 ± 0.5 -0.5 ± 0.6 -0.6 ± 0.4 OC\$2 -0.8 ± 0.6 -0.6 ± 0.5 0102
11
01 0.0 ± -3.0 ± 0.1 ± 0.0 ±
0.2 0.2 0.2 0.2
-0.2 -3.2 -0.1 -0.3
0.2 ± -2.6 ± 0.6 ± -0.6 ±
0.6 0.6 0.6 0.6
0.1 ± -2.7 ± 0.5 ± 0.7 ±
0.6 0.6 0.6 0.6
-0.2 -1.4 -0.5 -0.5
0.4 0.4 0.5 0.4
0.0 ± 0.6 -1.2 ±0.6 -0.3 ± 0.5 -0.6 ± 0.5
± 0.2 ±0.2 ±0.2 ± 0.2
± ± ± ±
the substantial differences in chemical perturbation of the heme sites and the ligands mimicking the bound state of Fell, the free energies of pairwise coupling show, with few exceptions, some definite and general trends. Unlike the case of CO binding, the oc\oc2 pair does not show appreciable coupling and therefore the a chains behave as independent in these model systems. The ft ft pair shows small but significant cooperativity and the same is seen for the ocx$2 pair. The a^ft pair is the most cooperative one in all systems. A striking observation is that in both the Fell/Felll-CN and CoIl/Fell-CO systems the free energies of pairwise coupling seem to be independent of the ligation state of the complementary pair. In the Fell/Mnlll system some differences are observed, most notably in the a1a2 pair that shows negative coupling at low saturation. In all cases, the pattern of pairwise coupling rules out the simplistic assumptions of the MWC model and echoes the results obtained in the case of CO binding. Direct nearest-neighbor interactions of the KNF-type dominate the scenario of site-site communication in these model systems also. Daugherty, Shea and Ackers (1994) have resolved all free energy levels for the Fell/Felll-CN systems as a function of pH. The free energies of pairwise coupling derived from their results are listed in Table 4.12 and sketched in Figure 4.14. Pairwise coupling is independent of ligation over the entire pH range 7.0-8.0. At pH > 8.0 pairwise coupling is strong at low saturation and tends to disappear at intermediate and high saturation.
4.4 Site-specific properties of hemoglobin
207
Table 4.12. Free energies ofpairwise coupling (in kcal/mol) for the Fell/ FeHI-CN hemoglobin system as a function ofpH (Daugherty et ah, 1994). Pair
0102
01022
01022
A022 OC\OC2
0C\$\
01022
01022
00 -0.3 -4.0 -0.2 -0.2 -0.3 -3.2
± 0.2 ± 0.3 ± 0.3 ± 0.3 ± 0.2 ± 0.2
-0.1 ±0.2 -0.2 ± 0.2 -0.9 ± 0.4 -2.7 ± 0.3 -0.7 ± 0.3 -0.7 ± 0.2 -1.1 ±0.3 -2.1 ± 0.4 -1.0 ±0.4 -1.0 ±0.3 -1.8 ±0.4 -2.4 ± 0.4 -1.6 ±0.4 -1.6 ±0.3 -2.1 ± 0.4 -2.5 ± 0.5 -1.8 ±0.4 -1.6 ±0.3
10
01
11
0.1 ± 0.3 -3.6 ± 0.3 0.2 ± 0.2 0.2 ± 0.3 0.0 ± 0.2 -2.9 ± 0.2 0.2 ± 0.2 0.0 ± 0.2 -0.4 ± 0.4 -2.2 ± 0.4 -0.2 ± 0.4 -0.3 ± 0.4 -1.6 ±0.4 -1.6 ±0.4 -0.5 ± 0.4 -0.5 ± 0.4 -0.5 ± 0.5 -1.1 ±0.5 -0.3 ± 0.5 -0.3 ± 0.5 -0.4 ± 0.5 -0.8 ± 0.5 -0.1 ± 0.5 -0.2 ± 0.5
0.1 ±0.3 -3.6 ± 0.3 0.2 ± 0.3 0.2 ± 0.3 0.0 ± 0.2 -3.0 ± 0.2 0.1 ±0.2 0.0 ± 0.2 -0.4 ± 0.4 -2.3 ± 0.3 -0.3 ± 0.3 -0.3 ± 0.4 -0.6 ± 0.4 -1.6 ±0.4 -0.5 ± 0.4 -0.5 ± 0.4 -0.5 ± 0.5 -1.1 ±0.4 -0.3 ± 0.4 -0.3 ± 0.5 -0.4 ± 0.5 -1.1 ±0.4 -0.4 ± 0.4 -0.2 ± 0.5
-0.5 ± 0.2 -4.2 ± 0.2 -0.4 ± 0.3 -0.4 ± 0.2 -0.2 ± 0.2 -3.2 ± 0.2 -0.1 ± 0.2 -0.3 ± 0.2 -0.2 ± 0.3 -2.1 ±0.4 -0.1 ± 0.4 -0.2 ± 0.4 -0.1 ± 0.3 -1.1 ±0.4 0.0 ± 0.4 0.0 ± 0.3 -0.3 ± 0.3 -0.9 ± 0.4 -0.1 ± 0.4 -0.1 ± 0.4 -0.1 ± 0.3 -0.8 ± 0.5 -0.1 ± 0.4 -0.2 ± 0.4
pH 7.0
pH 7.4
pH 8.0
pH 8.5
pH 8.8
pH 9.5
This would suggest that a concerted transition takes place after the first ligation step, thereby supporting an MWC-like mechanism of cooperativity at high pH. However, the behavior of the ax$x pair provides a notable exception. Coupling within this pair remains strong and positive even at high pH. In the complex hierarchy of site-site communications of the hemoglobin tetramer in this model system, the ocx$x and a2f52 dimers clearly stand out for the strength of cooperative coupling. The other pairs show a small coupling in the pH range 7.0-8.0, which, however, tends to increase significantly with pH. Also, the linkage between pairwise coupling and proton binding is eminently different for the ocx$\ and a2/32 dimers. The slope in the plots in Figure 4.14, divided by 2.303/?T, gives the net number of protons exchanged in the thermodynamic cycle relative to the pairwise coupling free energy. Protons are taken up as a result of
4 Specific cases and applications
208
O
q
J5
7.2
7.7
8.2
.8.7
9.2
9.7
q d
6.7
7.2
7.7
8.2
pH
8.7
9.2
9.7
4.4 Site-specific properties of hemoglobin
7.2
7.7
8.2
8.7
9.2
209
9.7
pH Figure 4.14 Free energies of pairwise coupling for hemoglobin pairs of chains, as indicated, as a function of pH. The values are listed in Table 4.12 and are computed from the free energy levels for the Fell/Felll-CN model system (Daugherty et ai, 1994). Symbols refer to different ligated configurations of the complementary pair as follows (see also Table 4.12): (open circles) 00; (open triangles) 10; (full triangles) 01; (full circles) 11.
210
4 Specific cases and applications
the interaction between ocx and ft, but are released in the interaction between ocx and a2, ocx and ft, or ft and ft. This result also deserves much consideration in view of the fact that the overall linkage with protons reconstructed from the data on the intermediates for the Fell/ Felll-CN system (Daugherty et al.9 1994) is practically identical to the Bohr effect of the native hemoglobin (Chu et al, 1982). The Bohr effect is the linkage between O 2 binding and proton release that allows hemoglobin to accomplish its important role in the blood (Bunn and Forget, 1986). The structural origin of the Bohr effect was intensely debated until the elegant work of Ho (1992) proved that the effect arises from the contribution of many ionizable groups. The findings by Daugherty et al (1994) suggest the existence of two classes of Bohr groups: one linked to the coupling within dimers and the other linked to the coupling between dimers. The foregoing analysis based on site-specific thermodynamics suggests that the Fell/Felll-CN system and its intermediates should be used in NMR studies to identify the Bohr groups directly. The importance of a site-specific analysis of hemoglobin stems also from the possibility of defining a mechanism for cooperativity from modelindependent considerations. Here we show how this can be done in practice. From the definitions given in Table 4.7 and the experimental results given in Tables 4.10-4.12, we assume that the free energies of coupling are independent of the ligation state of the complementary pair. This implies that the third-order and fourth-order interaction constants can be expressed in terms of the second-order interaction constants as follows caap = caacaPc'aP
(4.110a)
Capp = CppCapC'ap
(4.110b)
Caapp = CaaCppclpC*p
(4.110c)
The partition function for our model takes the form W(x) = 1 + 2(Ka + Kp)x + [caaK2a + cppK2p + 2{cap + c'ap)KaKp]x2 + 2capc'ap(caaKa + cppKp)KaKpX3 + caac ppclpc%K2aK2pX4 (4.111) The overall equilibrium constants written in terms of the model parameters are Ax=2(Ka + Kp) 2
A2 = caaK
tt
(4.112a) 2
+ cppK p + 2{cap + c'aP)KttKp
A3 = 2capc'afi(caaKa + cppKp)KaKp ^c%K\K\
(4.112b) (4.112c) (4.112d)
4.4 Site-specific properties of hemoglobin
211
The site-specific binding curves of the two chains are Xa = KaxXa/W(x) where Xa = 1 + [caaKa CC
+
C
+ (caf$ + c'aP)Kp\x
(4.113a) + ca(3c'al3(2caaKa
+
C
pp P 'pKKpX
Xp = KpxXp/W(x)
(4.113b)
where Xvv = 1 + [cppKp p p p + (c^^ + c ^ ) ^ ] * + This model easily accounts for the results obtained in the Fell/ Felll-CN system in the pH range 7.0-8.0. Also, it accounts for the high cooperativity in the global CO binding curve reconstructed from Perrella's results shown in Figure 4.10(a) and almost completely explains the distribution of intermediates for CO binding. Extension of the model can readily account for the results obtained with model systems at pH > 8.0. Pairwise coupling patterns can be used to formulate a revised two-state model where both the T and R states contain second-order interactions between chain pairs. Specifically, + 2(KaT [caaTKaT
+ CpprKpY
(4.114a) 2(KaR
(4.114b) (4.114c) \ JL
and all parameters are self-explanatory. The partition function for the symmetry rule (Ackers et ah, 1992) is embodied by eqs (4.114a)-(4.114c) as a special case, and so are several other models of hemoglobin cooperativity. Specific assumptions on the strength of pairwise interactions within either quaternary state can be made to simplify eq (4.114c) for practical applications. Models of cooperativity can be developed only after detailed information is obtained on the properties of individual sites and intermediates of
212
4 Specific cases and applications
ligation. The information contained in the global picture, say the O 2 or CO binding curve of hemoglobin, is minimal and many different interpretations yield equally acceptable results. In the local description, on the other hand, the ambiguity is resolved and only a few models survive the test of experimental data. It is noteworthy that the simple model based solely on pairwise coupling constants, given in eq (4.111), explains most of the results on the intermediates of ligation in model systems at pH <8.0 and the CO intermediates. In these systems, cooperativity is not dominated by a concerted transition between two states, as implied by the central idea of the MWC model (Monod et ai, 1965). Rather, the heme sites communicate directly in a pairwise fashion. Ligation at one site affects all other sites in contact with it through direct interactions. These ideas were first put forward by Pauling in 1935 (Pauling, 1935) and redefined in a more systematic way by Koshland et ah (1966). The existence of two distinct structures for the T and R states of hemoglobin (Perutz, 1970; Baldwin and Chothia, 1979; Shaanan, 1983; Fermi, Perutz and Shaanan, 1984) has widely been accepted, almost dogmatically, as the most direct demonstration of the validity of the MWC model for hemoglobin cooperativity. Detailed stereochemical and statistical thermodynamic mechanisms have been derived from analysis of these limiting intermediates of ligation (Perutz, 1970; Szabo and Karplus, 1972). More recently, new structures of fully-ligated hemoglobin have been resolved under low salt conditions (Smith, Lattman and Carter, 1991; Silva, Roger and Arnone, 1992; Smith and Simmons, 1994). The R2 structure that emerged from these studies is quite different from the R structure and is not intermediate on the pathway of the T-H> R transition (Janin and Wodak, 1993). Srinivasan and Rose (1994) have carried out a very thorough analysis of the T, R and R2 structures and demonstrated unequivocally that R is an intermediate between T and R2. The R structure is nothing but a 'handicapped' R2 structure, an artifact of crystallization conditions (high salt) that freeze the molecule in an intermediate state that does not exist under physiological salt conditions and prevents hemoglobin assuming its 'natural' R2 form. The interesting analysis by Srinivasan and Rose indicates that the detailed mechanisms of hemoglobin cooperativity based on the T and R structures may be flawed. Furthermore, if T and R2 are the two limiting states, the transition between them can take place through several intermediates that can be populated by direct site-site interactions, as predicted by the KNF model. The lesson to be learned from hemoglobin is far more general. Mechanisms of protein function inferred from crystal structures alone may be very
4.4 Site-specific properties of hemoglobin
213
misleading. It is not possible to understand mechanisms of cooperative transitions without gathering information on ligation intermediates under conditions that are relevant to physiological function (i.e., low salt). The landmark work of Perrella (Perrella and Rossi-Bernardi, 1981; Perrella et ah, 1983) and Ackers (Smith and Ackers, 1985) should be properly credited in this regard. Like the beautiful structural work of Perutz (1970), these functional studies have contributed enormously to our understanding of cooperative interactions in biology and will serve as a paradigm for any future studies on allosteric systems. We conclude our discussion of hemoglobin with the problem of the symmetry of the binding curve. It has been known for a long time that the O2 binding curve of human hemoglobin under physiological conditions is strongly asymmetric, since it becomes steeper at high saturation (Roughton et aL, 1955). Various attempts to relate this feature to a detailed mechanism of interaction among the chains have appeared in the literature (Peller, 1982; Weber, 1982). These approaches based on specific mechanisms are of limited validity. Site-specific thermodynamics has the advantage of being model-independent and the conclusions derived from this approach are valid in general. As shown in Section 2.3, symmetry of the binding curve X for hemoglobin demands A4 = (A3/Ai)2. This condition is independent of A2 and therefore symmetry, or lack of it, cannot depend on the interaction constants caa9 Cpp, cap or c'ap. Symmetry requires (Di Cera et al, 1992) Caaw(Ka + Kp)2 = (caaf>Ka + cafVKp)2
(4.115)
and depends on the association constants of the two chains and the thirdand fourth-order coupling constants. If Ka = Kp, asymmetry is observed whenever 4caapp =£ (caap + capp)2, which can be satisfied even if caap = capp, provided caapp =£ c2aap. Therefore, asymmetry of the O2 binding curve under physiological conditions demands neither subunit heterogeneity nor asymmetric interactions, contrary to the conclusions derived by Weber (1982) and Peller (1982). It is interesting to note the effect of symmetry on the site-specific binding curves Xa and Xp of the two chains. If these curves are symmetric, then one necessarily finds from eq (3.125) that the two centers of symmetry are (4.116a) (4.116b)
214
4 Specific cases and applications
These centers are identical to the mean ligand activities of the chains and therefore
= L = , = *m \[cK2K2
(4.117)
where xm is the mean ligand activity of the system as a whole. Hence, it follows from eqs (4.116a), (4.116b) and (4.117) that caapp = caapcapp. Substitution in eq (4.115) shows that asymmetry necessarily demands Caap ^ Capp, independent of the value of Ka and Kp. In the case of interest for hemoglobin, the left-hand side of eq (4.115) exceeds the right-hand side, which implies either (KjKp)2 > capp/caap> 1, or (Ka/Kp)2< under physiological conditions (Viggiano capp/caap< 1. Since KjKp^A and Ho, 1979), then capp/caap < 16 and also Xp > xa. The binding curve Xp is shifted to the right with respect to Xa. If Xa and Xp are not symmetric, there is no need to invoke asymmetric interactions to obtain a resulting global binding curve that is itself asymmetric. Knowledge of the shape of chain-specific oxygen binding curves under physiological conditions is therefore necessary to establish unequivocally whether the asymmetric oxygen binding curve of human hemoglobin is due to asymmetric interactions, or to other factors not accounted for by current mechanistic models. 4.5 Site-specific binding curves from global properties of structurally perturbed systems
If only global properties of a macromolecular system can be accessed experimentally, is it possible to derive relevant information on site-specific properties? This is a question of considerable importance in practical applications. In this section we show a possible approach to the problem, which combines the powerful tools of molecular biology with the principles of site-specific thermodynamics illustrated so far. This approach has the potential to broaden our understanding of cooperative phenomena in biology considerably by revealing the site-specific components of global effects from analysis of ad hoc perturbed systems. The practical advantage of this approach is obvious, since global properties of macromolecular systems are far easier to obtain experimentally than their site-specific components. We start with the basic relationship (Di Cera, 1990, 1994b)
'3^1 _ ! _ 3£> W()
W()
(4.H8)
4.5 Site-specific binding curves
215
The physical significance of eq (4.118) has been discussed at length. Here we are concerned with the operational significance of this equation. Note that the site-specific binding curve, Xj9 can be constructed from two global quantities: the partition function of the system as a whole, *F, and its contracted analogues OlP7, obtained when site ; is kept unligated, or llFy, obtained when site j is kept ligated. This holds true in general, regardless of the number of sites, or the cooperativity pattern. Hence, if a chemical perturbation of site / is applied in such a way as essentially to abolish binding to that site or to mimick its bound state, then information on the binding properties of site j can be derived. The perturbation acts as the source of information. Critical to the practical applicability of this approach is the assumption that the perturbation itself does not alter the network of interaction among the sites, nor does it change the binding affinities of the remaining sites. The assumption is tenable for simple systems. Indeed this strategy, cast in a different conceptual framework, was first introduced in 1895 by the Austrian chemist Wegscheider in the study of ionization reactions of polybasic acids (Wegscheider, 1895). Wegscheider's idea was to replace the ionizable carboxylate with a nonionizable group, an ethyl ester, that could mimick the protonated form of the 'wild-type' COO" group. By blocking carboxyl groups, ionization reactions at remaining groups could be studied in 'reduced' systems. Such 'reduced' systems are merely contracted forms of the system as a whole and are therefore amenable to description in terms of contracted partition functions. Neuberger (1936), among many others (Edsall and Blanchard, 1933), exploited Wegscheider's idea to resolve completely the three ionization reactions of glutamic acid into their site-specific components. The partition function for such a system is W(x) = 1 + (Kx + K2 + K3)x + (c12KxK2 + c13KxK3 + c23K2K3)x2 + c123KtK2K3x3 (4.119) where the suffixes refer to the ar-COO~ (1), NH2 (2) and y-COO" (3) groups, as indicated in Figure 4.15. The variable x denotes the proton activity. Neuberger synthesized esters of either the a- or y-carboxyl Site 1
Site 2
Site 3
COO-CHNH 2-CH2-CH2-COO" Figure 4.15 Schematic representation of glutamic acid in its fully deprotonated form. The three ionizable groups, or proton binding sites, are indicated.
216
4 Specific cases and applications Table 4.13. pK^s of glutamic acid and its ethyl esters (Neuberger, 1936). 3
pKa
Glutamic acid ar-Ethyl glutamate y-Ethyl glutamate Ethyl glutamate
2.155 3.846 2.148 7.035
4.324 7.838 9.19
9.960
groups and titrated the other groups. The results of his experiments are summarized in Table 4.13. The assumption embodied by the Wegscheider principle is that the ethyl ester effectively mimicks the protonated state of the carboxyl group. Hence, the data in Table 4.13 are de facto measurements of the equilibrium constants for the system as a whole and its contracted forms. The partition function for ar-ethyl glutamate is derived from eq (4.119) and Section 4.2 as follows ^ ( J C ) = 1 + (c12K2 + c13K3)x + c123K2K3x2 (4.120) In this derivative, proton binding to the amino and y-carboxyl groups is measured while the tf-carboxyl group is protonated. Likewise, the partition function for y-ethyl glutamate is ^3(x) = 1 + {cl3K, + c23K2)x + cn3KxK2x2 (4.121) Equations (4.119)-(4.121) are sufficient to resolve the site-specific parameters for glutamic acid. In the case of the 'wild-type' molecule we have (see Table 4.13) 109.960 = Ki
+ Ki
104.324+9.960 = CnKlK2 10 2.155 + 4.324 + 9.960 =
^
^
+ K3
+ c13KxK3 + c23K2K3 ^
^
(4.122a) (4.122b) (4.122c)
In the case of ar-ethyl glutamate we have 107.838 = CuKi
+ Ci3^3
103.846+7.838 = Cl23K2K3
(4.123a) (4.123b)
and for y-ethyl glutamate we have 10 919 = c13Kx + c23K2 1O2.i48+9.i9 = Cl23KxK2
(4.124a) (4.124b)
The resulting values of the site-specific parameters for the ionization reactions of glutamic acid are given in Table 4.14, along with their pK estimates. The results are identical to those obtained by Hill using a
4.5 Site-specific binding curves
217
Table 4.14. Site-specific parameters for proton binding to glutamic acid. K
pK
Site-specific binding constants (M" 1 ) Kx 5.69 x IO4 K2 9.12 x IO9 K3 1.26 x IO5 Interaction constants c12 ci 3 c23 cm
7.55 3.53 1.70 4.20
x x x x
10" 3 KT 1 lO" 1 10" 4
4.755 9.960 5.101 -2.122 -0.452 -0.770 -3.377
different approach (Hill, 1944). Note that the pKa of the amino group in ethyl glutamate can be predicted from the data in Table 4.14 to be 7.035, as found experimentally. In fact, titration of the amino group is expected to obey the second-order contracted partition function ( x ) = i + ±™-K 2x
(4.125)
Cl3
with a p # a given by pKa = log10 {cl22>c^K2). The site-specific analysis of glutamic acid yields useful information on the energetics of protonation from which structural information can be inferred. This point illustrates the importance of resolving site-specific energetics. When all groups are deprotonated, binding of the proton occurs with high affinity to the amino group and with low affinity to the carboxyl groups. However, the ar-carboxyl group has an affinity lower than the y-carboxyl group, due to the proximity of the amino group. The interaction constants are all less than one, indicating the presence of negative cooperativity in the protonation reactions. This effect acts in concert with the extreme heterogeneity of the sites to produce a strongly negatively cooperative proton binding curve for glutamic acid, as shown in Figure 4.16. Negative coupling among the sites is expected from electrostatic considerations. Again, the role of the amino group in differentially affecting the protonation properties of the two carboxyl groups should be emphasized. The two groups are located far enough away that c 1 3 « 1 when the amino group is deprotonated. Protonation of this group has an effect almost three orders of magnitude larger on the a- than the y-carboxyl group. Interestingly, the third-order interaction constant is practically the same as the product of all three second-order constants. This
4 Specific cases and applications
218
CO
d
co
d
CN
d
q d -14.0
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
-pH Figure 4.16 Proton binding curve of glutamic acid (continuous line) shown as fractional saturation, along with the site-specific proton binding curves (dashed lines) of the ar-carboxyl (right), ar-NH2 (left) and y-carboxyl (middle) groups. Curves were drawn using the parameter values listed in Table 4.14.
indicates that pairwise coupling in glutamic acid is direct, a result that echoes the hemoglobin scenario dealt with in the previous section. Can the Wegscheider principle be used in the case of complex macromolecular systems? Consider for instance the long standing problem of calculating the pKa of ionizable groups in proteins (Linderstr0m-Lang, 1924; Tanford and Kirkwood, 1957), where the site-specific proton binding curve of each ionizable group must be determined experimentally. For a protein containing twenty such groups, there are a total of 2 2 0 «10 6 configurations to be taken into account in the partition function \P and no experimental technique can provide such information. However, many residues like Asp, Glu, Arg, Lys and Tyr do not ionize in the pH range of physiological interest and can be treated as fully protonated or deprotonated. The remaining groups again form a 'reduced' system in the Weg-
4.5 Site-specific binding curves
219
scheider sense and the contracted partition function associated with them contains a drastically reduced number of configurations. This idea has been implemented in the analysis of site-specific titration curves and is known as the 'reduced-site model' (Bashford and Karplus, 1991). Given these assumptions, we can determine the site-specific titration curve of a His residue in a protein in the pH range of physiological interest as follows. First, the titration curve of the protein, X, is determined in the pH range of interest to obtain W. Here W denotes the partition function of all residues that ionize in the pH range under study. Then, a site-directed mutant of the His residue is made and the titration curve of the mutant is measured over the same pH range. The substitutions His-^Lys and His —> Val may be used to mimick the protonated and unprotonated states of the His residue. In the former case the titration curve of the mutant yields lXj (j labels the His residue), while in the latter it yields °Xj. These global binding curves yield the partition functions Anx A
I ^
(4.126a)
J—
Anx
ln°W;(jc) = I o ^ d l n x '
(4.126b)
J — oo
The partition function of the wild-type is derived from X as follows
\nW(x)= p X d l n * '
(4.127)
and application of eq (4.118) yields the site-specific proton binding curve for the His as follows f Anx
Xj=KjXexp I fXj-
1
f Anx
X)dlnx' = 1 - exp | (0Xj -
1
X)dlnx'\
(4.128) Integration of the difference titration curve gives a direct measure of Xj. Analysis of Xj reveals important properties of the system, e.g., whether the His is coupled to other residues and the nature of the coupling, positive or negative. The apparent pKa can be resolved from the mean ligand activity of the binding curve as 2303pKa = — lnx my. However, it may be reasonably objected that the His —> Lys or His —> Val substitutions may affect other properties of the macromolecule, i.e., the ionization of other groups, or the coupling of the His with other residues. In general, the assumption that a given substitution may actually mimick a particular ligation state for the His may be questioned. More importantly, how does this strategy apply to ligands such as metal ions, organic molecules,
220
4 Specific cases and applications
peptides and nucleic acids, for which binding sites in proteins involve large structural domains? For example, in the specific case of Ca2+ binding to an EF-hand (Kretsinger, 1980), what kind of substitutions are to be made to mimick the Ca2+-bound state of the site, or the Ca2+-free form? When binding of a ligand involves several protein residues, it becomes difficult to perturb the site as is done for proton binding reactions. In this case, the perturbation is likely to be more extensive and therefore the assumption that it mimicks a specific ligation state becomes untenable. We now show that site-specific binding curves can be obtained from eq (4.118) in an alternative way, using ad hoc perturbations of the binding affinity of the site of interest. This approach has a considerable practical advantage, since it is relatively easy to perturb binding sites structurally only slightly using recombinant DNA technology. The key advantage of our approach is that the requirement for the perturbation to mimick either the free or bound form of the site is no longer necessary. Also, it uses exclusively information derived from global properties of the system, thereby eliminating the need for directly monitoring site-specific effects at individual sites. Consider the partition function of the system as a whole written in terms of the first-order contracted partition functions involving site / W(x) = Ox¥j(x) + l^i{x)Kpc
(4.129)
Consider then a perturbation of site ;, say a chemical modification induced by a site-directed mutation, as illustrated schematically in Figure 4.17. The perturbation can either affect the properties of the site where it applies, or it can carry over to other sites. Assume that the perturbation remains localized at site ;, and therefore it affects only Kj. We speak in this case of a 'first-order perturbation' that changes the value of Kj to K\, while leaving the value of all other parameters unchanged. Clearly, the smaller the perturbation, the more likely that it will cause a first-order effect. In general, a first-order perturbation encompasses all cases where the change has its primary effect at the site where it applies and only a negligible secondary effect on other sites. Then, for the perturbed system we have \y(xy
= °\Pj(x)•+ 1Wj(x)Kfjx
(4.130)
The contracted partition functions are the same as those for the wild-type, or unperturbed system, since they do not depend on Kj. Given that both W and V are accessible experimentally as global properties of the system, the function
f = l - M . . == (i(i __ E\ E\Xj Xj J}
W(x)
\\
KjJ }} KJ
== mXj mXj ] J ''] J
(4.131)
K
}
4.5 Site-specific binding curves
221
Figure 4.17 Schematic representation of a perturbation applied to a specific site (bottom) of a biological macromolecule. The unperturbed system (top) binds two molecules at sites 1 (left) and 2 (right). The mutation at site 1 can either remain localized at this site (first-order perturbation), or carry over to site 2 (secondorder perturbation). If the perturbation is first-order, the site-specific properties of site 1 can be derived from knowledge of the partition functions W and W according to eq (4.131). can be constructed. Except for a constant factor rj ; , this function is the same as the quantity of interest Xj in the unperturbed, wild-type system. The constant factor rj; is easily derived from fi in the limit x —> °°. In the limiting case where mutation abolishes binding to site /, eq (4.131) yields eq (4.118). In general, eq (4.131) only requires rj; to be finite. The approach based on eq (4.131) should be tested with a number of mutants at site y, in order to guarantee the uniqueness of the function Xj. Different mutants may yield different perturbations of Kj9 but they should all give the same function Xj from eq (4.131). Only in this case are the perturbations at site j truly first-order. A simple way to implement this strategy is to construct the 'doubleratio' plot in Figure 4.18. Once the partition function is resolved in the global description for the wild-type and a number of mutants, the overall equilibrium constants As can be derived. The first and last such constants are of particular importance. In fact, A x = K x + K 2 + ... K j . . . + KN
(4.132a)
A[ = Kx + K2 + . . . K'j . . . + KN
(4.132b)
A
(4.132C)
A-N
N
= c 1 2 . . .,-.. , N K X K 2 ... K J . . . K = C\2
mj %
jyKiK2
N
... Ay ... A w
\f+.L5ZQ.)
222
4 Specific cases and applications
Low-affinity site
in
d
High-affinity site
o d 0.0
0.5
1.0
1.5
A'N/AN Figure 4.18 Double-ratio plot for the analysis of first-order perturbations. The values of A7v for the mutants, relative to AN for the wild-type, are plotted versus the ratio A[/A\. Two limiting cases in this plot denote perturbation of a high-affinity (slope = 1), or low-affinity (slope = 0) site. Points outside these boundaries denote violations of the first-order perturbation hypothesis. Points following a straight line inside the boundaries support the validity of the firstorder perturbation hypothesis.
where the prime refers to the mutant of site j . Hence, A[ Ki + K2 + . . . + KN k K2
KN A
N
(4.133) If several mutants perturbing site j are analyzed and the results plotted, a straight line should be observed in the double-ratio plot if the first-order approximation is obeyed. The relationship (4.133) deserves consideration. The intercept and slope are correlated and sum to one. The slope, p, is bounded from zero to one and reflects the value of Kj in the unperturbed
4.5 Site-specific binding curves
223
system relative to Ax. The intercept, 1 — p, is likewise bounded from zero to one and reflects the value of the sum of all site-specific binding constants, but Kj9 relative to Ax. The first-order approximation imposes a constraint on the double-ratio plot. Two boundaries define the limiting behavior of the system in response to the perturbation. The boundary with zero slope defines the case in which AN is perturbed with practically no change in Ax. This case applies when Kj is negligibly small compared to all other Ks and the site being perturbed has low affinity. The boundary with unit slope defines the case in which Ax changes linearly with AN. This is the case in which Kj is large compared to all other Ks and the site being perturbed has high affinity. The two boundaries cross over at the point reflecting the wild-type behavior. The necessary condition for the firstorder approximation to be valid is that the experimental points lie within the foregoing boundaries, as illustrated in Figure 4.18. The condition is also sufficient if data points are within the boundaries and obey eq (4.133). In this case, the value of Kj can be derived directly from the slope in the plot. If points lie outside the boundaries, then they violate the first-order approximation. This case provides a direct and simple demonstration that the mutation has carried over to other sites. The double-ratio plot is therefore extremely informative. Assume, then, that the perturbation at site j carries over to other sites. We speak in this case of second-order, third-order or nth-order perturbation, depending on how many sites are involved. The approach based on eq (4.131) is easily generalized. For example, in the case of a second-order perturbation the partition function of the wild-type is written in terms of second-order contracted partition functions as follows 10 xy(x) = °°*p W^ + iy(jc) + (
01x
¥ijKj)x + nWijcijKiKjx2
(4.134)
If the mutation affects Kt and Kj, but leaves all other site-specific terms unaltered, then + nWijCijKfiKfjx2 (4.135) xp(xy = °°^.(x) +(^WijKi + ^ijK^x The function ftj, analogous to fi in the case of a first-order perturbation, in this case yields (x) = TIM + rijXj - 'WirijXiXj
(4.136)
where
_ VW
( 4 . 1 3 7)
224
4 Specific cases and applications
is the pairwise coupling function between sites i and /, and the rys have the same form as in eq (4.131). As expected, the function ftj contains terms relative to sites / and /, along with the coupling between them. The interaction constant ctj can be derived in the limit x —> °° once ^ and ;r/ are known from first-order perturbation experiments, where Kt and Kj are perturbed separately. Likewise, measurements of W and V can be used to access 11WIy using eqs (4.136) and (4.137) and to resolve all the overall binding constants of a second-order contracted partition function of the wild-type system. Similar arguments apply to higher-order perturbations, or those cases where interaction constants can also be affected. We now show an application of the foregoing approach to the analysis of cooperative Ca2+ binding to calbindin, one of the smallest members of the calmodulin superfamily (Skelton et ah, 1994). Calbindin is composed of two helix-loop-helix motifs responsible for Ca2+ binding, as shown in Figure 4.19. The C-terminal site (site 2) has the amino acid sequence and fold of an archetypal EF-hand, while the N-terminal site (site 1) differs from the usual EF-hand in that it contains two additional residues (Szebenyi and Moffat, 1986). The protein binds Ca2+ with positive
Figure 4.19 Ribbon representation of calbindin, as derived from the crystal structure (Szebenyi and Moffat, 1986). The bound Ca2+ ions are shown by circles. Site 1, shaped as a pseudo EF-hand, is at right. Site 2, shaped as an archetypal EF-hand, is at left.
4.5 Site-specific binding curves
225
cooperativity and a Hill coefficient of about 1.5 (Linse et aL, 1987, 1988, 1991). Due to the structural differences between the binding sites, it is expected that the true coupling between them is partially reduced by site heterogeneity. The arguments discussed in Section 3.1 become particularly relevant in this respect. In fact, calbindin provides a sort of 'hydrogen atom' for site-specific thermodynamics, being the simplest cooperative system with site heterogeneity. Experimental evidence provided by NMR suggests that Cd2+ binds with higher affinity to site 2 (Akke, Forsen and Chazin, 1991) and that the binding pathways [00] -» [10] -> [11] and [00] -+ [01] -» [11] induce distinct structural transitions in the molecule (Carlstrom and Chazin, 1993). Spectroscopic and kinetic studies on the global Ca2+ binding properties of calbindin suggest that site heterogeneity may be a factor of 4-6 (Linse et aL, 1987, 1991; Brodin et aL, 1990; Martin et aL, 1990). However, no direct evidence has been provided for Ca2+ binding with higher affinity to site 1 or site 2. Electrostatic calculations suggest that site 2 may have only a slightly higher affinity (Ahlstrom et aL, 1989). Valence maps indicate that site 1, rather than site 2, may bind Ca2+ with higher affinity (Nayal and Di Cera, 1994). Furthermore, these maps suggest that Cd2+ should not be used as a model for the energetics of Ca2+ binding, since the parameters relating bond strength to bond order are significantly different for these metal ions (Pauling, 1929, 1947; Brown and Wu, 1976). A number of mutants of residues in and around site 1, and partially site 2, have been made to assess the role of electrostatic contributions to the binding of Ca2+ and the global binding parameters have been resolved for all of them (Linse et aL, 1987, 1988, 1991; Martin et aL, 1990). Some of the residues mutated in the wild-type are shown in Figure 4.20. Particularly interesting is the observation that mutations around site 1 remain localized at this site and do not propagate to site 2 (Linse et aL, 1987; Martin et aL, 1990). The global binding parameters for the wild-type calbindin and a number of mutants are summarized in Table 4.15. Mutations significantly affect the binding affinity and cooperativity. We will use this set of mutants to resolve the site-specific parameters for calbindin. We assume that the single mutants around site 1 provide a small perturbation of the affinity of site 1, while leaving the affinity of site 2 and the interaction between the sites unchanged. Our assumption of the first-order perturbation is supported by NMR data showing that the mutants E17Q, E26Q and E60Q have practically the same structure as the wild-type protein (Akke et aL, 1991; Carlstrom and Chazin, 1993). The mutant P20G and the deletion mutants P20G, AN21 and AP20 show
226
4 Specific cases and applications
Table 4.15. Global binding parameters for Ca2+ binding to calbindin and its mutants (Linse etal, 1987, 1988, 1991). (IVI-1) Wild-type P20G P20G, AN21 AP20 Y13F E17Q D19N E26Q E60Q E17Q/D19N E17Q/E26Q D19N/E26Q E17Q/D19N/E26Q
2.0 6.5 1.7 8.0 3.7 2.5 4.0 6.3 1.0 2.5 6.3 7.9 6.3
X X X X X X X X X X X X X
108 107 108 107 108 107 1077 10 108 107 106 106 106
2 A2 CM" )
k! (M - 1 )
k2 (M - 1 )
1016 1014 1013 1013 1016 1015 1015 1016 1016 1014 1014 1014 1013
1.0 X 108 3.3 x 107 8.5 X 107 4.0 X 107 1.9 X 108 1.3 X 107 2.0 X 1077 3.2 X 10 5.0 X 107 1.3 X 107 3.2 X 106 4.0 X 106 3.2 X 106
7.9 1.6 1.0 1.0 4.9 2.5 2.0 5.0 6.4 2.5 1.3 8.0 1.0
7.9 5.2 8.5 4.0 9.3 3.2 4.0 1.6 3.2 3.2 4.0 3.2 3.2
X X X X X X X X X X X X X
X X X X X X X X X X X X X
108 107 106 106 108 108 1088 10 108 107 108 107 107
1.5 0.8 0.2 0.3 1.2 1.6 1.5 1.6 1.6 1.2 1.7 1.6 1.3
D19 E26
E17 Figure 4.20 Ribbon representation of sites 1 (bottom) and 2 (top) of calbindin, indicating the side chains of residues whose mutations obey the first-order perturbation hypothesis. The bound Ca 2+ ions are shown by circles.
drastic perturbation of the macroscopic cooperativity pattern. Nonetheless, there is evidence that the structural perturbation is confined to site 1 (Linse et al.9 1987). The perturbation of the global binding constants in these mutants may indeed reflect a perturbation of the site-specific binding
4.5 Site-specific binding curves
227
constant Kx only. Construction of the double-ratio plot in Figure 4.21 confirms this expectation. The mutant Y13F clearly lies outside the boundaries for first-order perturbation. Replacement of Tyrl3 with a Phe induces changes that propagate to site 2 and may also involve the communication between the sites. Other mutations lie within the boundaries, but do not give rise to a linear dependence of the perturbed binding constants. On the other hand, there are five mutants whose perturbed constants clearly follow a straight line in the double-ratio plot. The values of Ku K2 and c12 are readily derived from this plot, since the ratios KxjAx
LT)
O
q d 0.0
0.5
1.0
1.5
An I A , Figure 4.21 Double-ratio plot for calbindin. The point outside the boundaries refers to the mutant Y13F, which clearly violates the first-order perturbation hypothesis. The results pertaining to five mutants (filled circles) follow a straight line according to eq (4.133). These mutants are given in the key on Figure 4.22. The point of coordinates 1,1 refers to the wild-type. Since the straight line is closer to the high-affinity boundary (see also Figure 4.18), mutations affect predominantly the high-affinity site 1. The value of K\ is derived from the value of Ax and the slope in the plot and is 1.7 x 108 M"1. The values of K2 and cu are 2.7 x 107 M"1 and 18.
228
4 Specific cases and applications
and K2/A1 are given by the slope and intercept respectively, while Ax and A2 are known. From the results in Figure 4.21 it also follows that the site being perturbed, site 1, is the high-affinity one since the straight line is closer to the high-affinity boundary (see also Figure 4.18). A striking observation is that the mutants behaving according to the first-order perturbation are all isosteric substitutions of negatively charged residues around site 1. The site-specific binding curves of the two sites can be reconstructed from application of eq (4.131). The partition function of unperturbed calbindin is (4.138) W(x) = 1 + Axx + A2x2 = 1 + (KX + K2)x + cl2KxK2x2 For the mutant obeying first-order perturbation of site 1 we have W{x)' = 1 + A[x + A'2x2 = 1 + (K[ + K2)x + cuK[K2x2 (4.139) Application of eq (4.131) yields Xu and X2 is derived from the global binding curve of the wild-type as X2 = X — Xx. The results are shown by symbols in Figure 4.22 for the five mutants on the straight line of Figure 4.21. The predicted site-specific binding curves are practically identical for all these mutants, although the perturbation of the overall binding constants is quite different in each case, as shown in Table 4.15. Site 1 binds Ca2+ with an affinity of about 1.7 x 108 M"1, which is nearly six times the affinity of site 2. As a result of cooperative coupling between the sites, the affinity of either site increases by a factor of eighteen when the other site is ligated. This analysis provides a solution for the site-specific problem for calbindin and indicates that site 1 is probably the high-affinity binding site, contrary to the conclusion drawn from Cd2+ binding studies (Akke et al., 1991). The results are also consistent with the structure of calbindin. The crystal structure of site 1 shown in Figure 4.20 suggests that it is unlikely that a mutation of Glu26 will be carried over to site 2, and the same applies to Glul7. These residues are about 10 A closer to the Ca2+ in site 1 than that in site 2. Aspl9 and Glu60 are close to both sites and, in principle, mutations of these residues should perturb sites 1 and 2. In practice, however, mutation of these residues yields effects similar to mutation of Glu26 and Glul7. A molecular dynamics simulation of calbindin shows that Glu60 may actually be part of the coordination sphere of Ca2+ in site 1 (Ahlstrom et al., 1989). This residue, and possibly Aspl9, may be closer to site 1 in solution, contrary to the conclusions drawn from the crystal structure (Szebenyi and Moffat, 1986). It may also be hypothesized that these residues may be more strongly coupled to site 1 than site 2 so that although perturbation of site 2 cannot be ruled out in principle, it is likely to be insignificant in practice.
4.5 Site-specific binding curves ^
229
I
O • * • •
00
d
E17Q D19N E26Q E60Q E17Q/D19N
O
3
q d -10.0
-9.5
-9.0
-8.5
-8.0
-7.5
-7.0
log [Ca 2+ ] (M) Figure 4.22 Site-specific Ca2+ binding curves of site 1 (left) and site 2 (right) of calbindin, as derived from the global binding parameters of the wild-type and the mutants as listed. Curves were drawn from eqs (4.27a) and (4.27b) using parameter values: ^ = 1.7 x 108 M"1, K2 = 2.7 x 107 M"1 and c12 = 18.
For a two-site macromolecule, knowledge of the site-specific binding curve of either site and the overall binding constants is sufficient to resolve all site-specific binding parameters. In the case of calbindin these parameters can be derived directly in the first-order perturbation using the following transformations of the overall binding constants A.
A ' _
A
A.
A'2-A2 A'2A, - A[A2 Af2-A2 A2
(4.140a) (4.140b)
ci2=^r(4.140c) The results are listed in Table 4.16 for all mutants. The mutants on the
230
4 Specific cases and applications Table 4.16. Site-specific binding parameters for Ca2+ binding to calbindin. Derived from
Kx
P20G P20G, AN21 AP20 Y13F E17Q D19N E26Q E60Q E17Q/D19N E17Q/E26Q D19N/E26Q E17Q/D19N/E26Q
1.4 4.1 1.2 7.7 1.8 1.7 1.7 1.7 1.8 1.9 1.9 1.9
X X X X X X X X X X X X
I-1)
K2
108 107 108 108 108 108 108 1088 10 108 108 108
6.2 1.6 7.9
X X X
107 108 107
9 12 8
1.8 3.1 2.9 3.4 2.4 5.3 7.2 6.2
X X X X X X X X
107 107 107 1077 10 106 106 106
24 15 16 14 19 77 58 66
(
*
•
I-1)
straight line of Figure 4.21 yield very similar estimates for the site-specific parameters, as expected. These estimates differ slightly for the mutants P20G and AP20 that are also in close proximity to site 1. However, the difference may not be statistically significant, since the error on the global parameters reported for these mutants is large (Linse et al.9 1987). The double and triple mutants, on the other hand, provide very different estimates for the site-specific parameters, and so do the mutants P20G, AN21 and Y13F. The double mutant E17Q, D19N provides a notable exception. It may be concluded that neutralization of two or three surface charges induces changes that propagate to site 2 and possibly affect the interaction between the sites as well. Deletion of Gln21 may drastically affect the overall conformation of calbindin and the substitution Y13F clearly affects Kx and K2, as well as c12, since Ax increases while A2 stays constant (see also Figure 4.18). No physically meaningful values for K2 and c12 can be derived from this mutant in the first-order perturbation approach, since eq (4.140b) returns a negative value. 4.6 Pairwise coupling
We now focus on the concept of 'pairwise coupling' already introduced in Section 4.4. This concept has an applicability that goes well beyond site-specific binding cooperativity and extends to practically all cooperative processes that can be accessed experimentally in terms of site-specific effects. The relevance of this concept stems from the possibility of
4.6 Pairwise coupling
231
'breaking' the code for cooperativity in a model-independent fashion. This is an aspect of central importance to a variety of processes as diverse as ligand binding, protein folding and mutational perturbations. All these processes can be cast in terms of an underlying conceptual framework based on site-specific thermodynamics. The advantage of this approach is that the specific details of the system under investigation become of marginal interest, since the energetics are treated in their most general form. Analysis of a cooperative process involving a macromolecular system entails the following three steps: (1) Identification of structural domains important for energetic coupling. (2) Assessment of the coupling among different domains. (3) Elucidation of the precise mechanism of coupling.
These steps encapsulate the relevant aspects of the problem and its solution, regardless of the specific process under study, whether it involves ligand binding, protein folding or mutational perturbations. The first step determines the 'degrees of freedom' of the system, like the number of binding sites in the case of ligand binding. In the case of protein folding studies, binding sites are replaced by folding domains, like helices and sheets, that can be treated as elementary units. In mutational perturbation studies, the first step is equivalent to identifying those residues whose mutation significantly affects the energetics under investigation, e.g., substrate specificity or protein stability. The second step establishes a connection among sites, domains or residues from analysis of coupling effects. Examples are given by site-site communication leading to cooperative ligand binding, coupling of helices and sheets that determines the correct folding pattern, or coupling of individual residues as revealed by double mutations. The third step is the most important and challenging, since it provides a direct connection between structure and energetics. Once residues, sites or domains important for protein function are identified, along with interaction patterns, the key problem is to decipher the exact mechanism of communication among individual elements. Such mechanisms are often assumed a posteriori and modeled to fit the specific properties of the system. Site-specific thermodynamics provides a modelindependent strategy to 'break' cooperativity codes that can be used in general, without formulation of ad hoc assumptions. This strategy is based on the properties of the coupling free energy in a thermodynamic cycle. A cooperative system composed of N elements, like binding sites, protein domains or residues, can be associated with an Af-dimensional
232
4 Specific cases and applications
vector of binary digits, 0 and 1. The entries 0 and 1 stand for free and bound in the case of binding sites, unfolded and folded in the case of protein domains, or wild-type and mutated in the case of mutational perturbations. There are 2N possible configurations for the system, as already pointed out in the case of ligand binding. Pairwise coupling between any two elements, / and /, is defined as the coupling free energy in the thermodynamic cycle involving the 0 -»1 switching reactions of these elements, as illustrated in Figure 4.23. Keeping the remaining N — 2 elements in a given configuration, the cycle in Figure 4.23 has a coupling free energy (4.141) AGC = 1AGi - °AG, =xAGj - °AGy The elements are positively linked if the coupling free energy is negative, and negatively linked otherwise. The case AGC = 0 indicates the absence of coupling. Since there are N — 2 other elements, besides i and j , then there must be a total of 2N~2 possible configurations for which the coupling between / and j can be computed. Hence, the total number of coupling free energies in the system is v = ( N \2N~2 = N(N - l)2iV"3
(4.142)
The binomial term enumerates the possible ways of selecting any two elements among N, while the power law term reflects the number of thermodynamic cycles that can be constructed for any given pair of elements by exhausting the configurations of all other N — 2 elements. There is obviously no coupling free energy for N = 1 and only one value for TV = 2, as expected. The value of v increases rapidly with N. The approach based on the thermodynamic cycle in Figure 4.23 has been widely used in the analysis of mutational perturbations (Ackers and
AGt
oo °AGj 0 O# l
AGt
Figure 4.23 Thermodynamic cycle for the coupling between elements / and /. Symbols denote states 0 (open circles) and 1 (filled circles). The free energy of coupling between / and j is given by eq (4.141).
4.6 Pairwise coupling
233
Smith, 1985; Horovitz and Fersht, 1990; Wells, 1990) and binding cooperativity (Jencks, 1981; Di Cera, 1990). Here we extend previous approaches by drawing attention to an important property of the cycle. Once coupling free energies are calculated for all possible configurations of the system, it is possible to decipher the code for cooperativity using the following properties of pairwise coupling: Property 1 If the coupling interactions involving two elements are direct, the free energy of pairwise coupling is independent of the configuration of the remaining N — 2 elements. Property 2 If the coupling interactions involving two elements are indirect, the free energy of pairwise coupling is dependent on the configuration of the remaining N — 2 elements.
We now prove the first property in a rigorous manner. Consider the elements i and j and assume that they are coupled directly to each other and to all other elements. This guarantees that the coupling between i and j is not mediated by indirect interactions. To compute the coupling free energy in the thermodynamic cycle in Figure 4.23 for an arbitrary configuration of the other N - 2 elements, we have to evaluate the contribution of all interactions to the free energy terms. When i and j are in state 0, the free energy associated with the system is N
Li
Mi
Lj
'/Goo = G,o + Gh0 + X G/,e + '^oo + 2 ilUoo + £ ^01 +A
(4.143)
The terms in the foregoing expression refer to the free energy of elements i and j in state 0, Gifi and Gjt09 the free energy of element / =£ /, / whether it is in state 0 or 1, G/,©, the interaction free energy between / and j when both are in state 0, i]'Um9 and the interaction free energies between / and any other element, and likewise for / and any other element. These interaction terms can be partitioned into two groups, one containing all elements other than i and j in state 0 and the other containing all elements other than i and j in state 1. There are Lt elements in state 0 and Mt elements in state 1 that interact with i. Likewise, there are L ; elements in state 0 and M; elements in state 1 that interact with /. The term A contains all interaction terms among the N - 2 elements other than / and j . It is inconsequential to enumerate these terms, as well as to specify whether the interactions are direct or indirect, since the term A is by definition independent of the state of elements i and j and hence makes no
234
4 Specific cases and applications
contribution to the coupling free energy between these elements. When / switches to state 1, the free energy of the system is Pi
N
ij
Qi
a
*G10 = G u + Gjfl + X G/,e + Uio + 2 Ul0 + £
i!
Un
+ 2*^00+2*^01 +A l*i,j
(4.144)
l±i,j
The free energy now contains new terms. Specifically, the free energy of element / in state 1, G u , the interaction term between i and j with / in state 1 and j in state 0, ijU10, and the interaction terms between i and all other elements with i in state 1. Again, these terms are partitioned into two groups, Pi involving elements in state 0 and Qt involving elements in state 1. Clearly, L t = P{
(4.145a)
Mt = Qt
(4.145b)
because the number of direct interactions involving i is fixed. The free energies pertaining to all elements other than / and j remain unchanged, and so are the interaction terms for element /. Likewise, when j switches to state 1, the free energy of the system is Li
N
ij
Mi
Pj
G01 = G,-,o + Ghl + 2 G ; , e + '^01 + £ l7£/oo + 2 ' U i 2
+A
(4.146)
with Lj = Pj
(4.147a)
Mj = Qj
(4.147b)
Finally, when both i and ; are in state 1, the free energy is N
Pi
<>GU = G u + G}tl + 2 G/,© + liUn + 2«£/ 1 0 + ^lUlt
+A
ft
^aUu
P,
(4.148)
The coupling free energy in the thermodynamic cycle in Figure 4.23 is AGC = 'Ad
ij y - °AG,- = 'AGj - °AG y = "Goo + Gn - *"G10 - ' G01
(4.149)
4.6 Pairwise coupling
235
Substitution of eqs (4.143), (4.144), (4.146) and (4.148) into eq (4.149) yields AG C = f''£/oo + ijUn ~ ijU10 - i]'U01
(4.150)
The foregoing expression is independent of the particular state of elements other than i and /. Hence, if the coupling interactions involving i and j arc direct, the coupling free energy is independent of the configuration of all other elements. This result is general and holds regardless of the specific nature of the direct interactions involving i and /. Also, it is independent of the nature of the interaction among all other elements, whether they are direct or indirect. The second property is easier to prove. It is intuitively obvious that if the coupling between i and j is indirect, then there must be other elements responsible for the linkage and therefore the coupling free energy will necessarily depend on the state of these elements. The free energy of the system in the presence of indirect interactions, when i and j are in state 0, is given by ^oo = Gi>0 + G/t0 + Uoo(o)
(4.151)
where Um(o) is the interaction between i and j , given the particular configuration o of the other N — 2 elements. The other free energy terms in the thermodynamic cycle in Figure 4.23 are given by ij
G10 = Gifl + G,.,o + U10(o)
(4.152)
G01 = G it0 + Gul + U01(o)
(4.153)
UGn = Gul + G/tl + Un(o)
(4.154)
ij
Hence, AGC = Uoo(o) + Un{o) - U10(o) - U01(o)
(4.155)
and the coupling free energy clearly depends on the configuration of the elements other than i and j . A specific example is provided by a two-state model where each element can exist in two possible energetic states, T and R. It is assumed that the T-> R transition is positively linked to the 0-» 1 transition, and that the transition is global and affects all elements in a concerted fashion. This is essentially a generalization of the MWC model (Monod et ai, 1965) that allows for treatment of cooperative processes other than ligand binding, like protein folding or mutational perturbations. The various free energy terms involved in the thermodynamic cycle in Figure 4.23 for such
236
4 Specific cases and applications
a generalized model are = RG,-0 + RGy;0 +
N
G
>exp .l=ti,j\
RT In
'
1 1 —1—
1
Kl
(4. 156a)
L +1 N
R
R
= G U -I" G,,o + L
G
Lc u c, ; O exp| L
RT In
'
1 1 —1—
TfT
1
Kl
(4- 156b)
L +1 N
R
= G,-,0 -•- Gy,i + R
L
G
RT
'
cxp|
In
1 1 —1—
1
(4 .156c)
L +l N
R
*• Gj,i
+ r•
- i?Tln
^ •v i
Lcitlcp i exp
I
/ -n 1
'G /,©
L + l
~
T<
^
'
1 1 —1—
1 (4. 156d)
where the R Gs are the free energy levels for state 0 in the R state, RG/>© and TG;© denote the free energy levels for states 0 or 1 in R and T for the elements other than i and / , L is the allosteric constant reflecting the relative concentrations of T and R when all elements are in state 0, and (4.157a)
(4.157b)
(4.157c)
4.6 Pairwise coupling
237
—-^
^M RT ) The free energy of pairwise coupling for this model is
(4.157d)
AGC = -RT In
(4.158) It is straightforward to prove that eq (4.158) is always negative, thereby implying that the coupling between any two elements in the system is always positive. In fact, eq (4.158) can be rewritten in the simpler form AGC = -RTln * + ( W 1 (
+ Ci )fL
^
+ C
)/L
^C^fL
f2L2
(4.159)
where / replaces the exponential term. The numerator and denominator of eqs (4.158) and (4.159) are second-order polynomials in L that differ only in the coefficient of L. Simple transformations prove that Ci,oCj,o + cuxcux
> culcjS> + C/,0^,1
(4.160)
since (ci,i - ci9o)(cJ9l - cuo) > 0 (4.161) This is because, by hypothesis, the T—> R transition is positively linked to the 0—> 1 transition. If the R state has a lower free energy than the T state, then state 1 must have a lower free energy than state 0. Hence, c u is necessarily less than one, but is also smaller than c /0 , because it is more likely to find element / in the T state when it is in state 0. The same applies to element /, so that eq (4.161) is satisfied. Complex patterns of coupling free energies can be obtained in this model if each element shows a different linkage between the T—> R and 0—> 1 transitions. In general, the free energy of coupling will depend on the state of the other elements, contrary to the case of direct interactions in eq (4.150). Pairwise coupling profiles can be constructed for a variety of cooperative processes. An application to ligand binding is given in Section 4.4. The approach is general enough to be extended to other relevant processes, like protein folding and mutational perturbations. Mutational perturbations are discussed in some detail, due to the widespread use of sitedirected mutagenesis studies in the analysis of structure-function relation-
238
4 Specific cases and applications
ships in macromolecules (Knowles, 1987). Mutational perturbations are currently used in a variety of systems to explore folding pathways (Horovitz and Fersht, 1992), protein stability (Lyu, Liff, Marky and Kallenbach, 1990; Padmanabhan et ah, 1990; Matthews, 1993), enzyme specificity (Carter and Wells, 1988; Scrutton, Berry and Perham, 1990; Wells, 1990; Cornish and Schultz, 1994) and molecular recognition (Ebright, 1986; Lesser, Kurpiewski and Jen-Jacobson, 1991; Draper, 1993). In most of these studies early events of protein folding or important aspects of protein stability have been dissected using single and double mutations. The approach based on pairwise coupling requires the construction of triple, quadruple and higher-order mutants. For the approach to be exploited fully, it is necessary to construct a complete set of mutations that includes at least three elements. This is because the number of thermodynamic cycles for pairwise coupling must be greater than one, which is seen if and only if N ^ 3 in eq (4.142). Hence, mechanisms of cooperativity cannot be dissected in a model-independent fashion using only single and double mutations. Assume that the problem at hand is to determine the mechanism through which the four helices A, D, F and H of apomyoglobin communicate and give rise to protein stability. First, the free energy of unfolding is derived for the wild-type. This is the quantity to be used in the thermodynamic cycle in Figure 4.23. Second, single site-directed mutations are made in each of the helices, as shown in Figure 4.24, to identify residues that perturb the stability of apomyoglobin. Third, once the best candidate is identified for each helix from single mutations, the entire set of mutants up to the quadruple mutant is constructed. This set can be depicted in vectorial form as [0000] [1000] [0100] [0010] [0001] [1100] [1010] [1001] [0110] [0101] [0011] [1110] [1101] [1011] [0111] [1111]
(4.162)
and is identical to the ligand binding case dealt with in the scheme (4.72). The vector [0000] denotes the wild-type, [1000], [0100], [0010] and [0001] pertain to single mutants on helices A, D, F and H, and so forth for the double, triple and quadruple mutants. The pairwise coupling pattern is constructed by measuring the free energy of unfolding for all mutants. This yields the stability of the four species in Figure 4.23 and hence the
4.6 Pairwise coupling
239
Figure 4.24 Ribbon representation of apomyoglobin. Circles depict residues on the A, D, F and H helices that are assumed to be critical for protein stability.
coupling free energy from the general eq (4.149). There are a total of twenty-four such values and analysis of the pattern reveals whether the four helices are coupled directly or indirectly. The same approach can be used to assess the mechanism of coupling within each structural element. For example, the stability of helix A can be studied by multiple substitutions within the helix and subsequent analysis of the pairwise coupling pattern. As an example of the approach based on pairwise coupling we analyze the mutations of the salt-bridged triad Asp8, Aspl2 and ArgllO of barnase (Horovitz et aL, 1990, 1991; Horovitz and Fersht, 1992). All the
4 Specific cases and applications
240
Table 4.17. Free energies of pairwise coupling (in kcal/mol) for the stability of barnase (Horovitz and Fersht, 1992). Pair Asp8-Aspl2 Asp8-ArgllO Aspl2-ArgllO
0
1
-0.44 ± 0.06 -0.98 ± 0.06 -1.25 ± 0.06
0.33 ± 0.06 -0.21 ± 0.06 -0.48 ± 0.06
D8D12R110 0.00±0.03 ADR 0.99+0.03
DAR 0.34±0.03
DDA 0.46±0.03
AAR 0.89+0.03
ADA 0.47+0.03
DAA -0.45 ±0.03
AAA -0.11+0.03
Figure 4.25 Schematic representation of the complete set of mutants of the salt-bridged triad Asp8, Aspl2 and ArgllO of barnase (Horovitz and Fersht, 1992). The free energy of unfolding (in kcal/mol) relative to the wild-type, as derived from eq (4.163), is shown for each mutant. The pairwise coupling patterns derived from the set of mutations are given in Table 4.17.
residues of this triad have been mutated to Ala in a complete set and the free energy of unfolding has been measured for all members of the set. The results are given in Figure 4.25, as calculated relative to the wild-type using the expression (4.163) AAGU = wtAGu - 'AG,, where AAGU is the difference in the unfolding free energy between the wild-type and the mutant. A positive value of this parameter indicates that the mutant is less stable than the wild-type and vice versa for negative values. The pairwise coupling patterns for each of the three pairs are listed in Table 4.17. It is evident from the table that the three residues are indirectly linked in the wild-type and contribute to the stability of the protein as a 'unit'. Interestingly, mutations tend to interact with positive cooperativity in the stabilization of the protein. The close proximity of the residues explains their behavior as a 'concerted switch', so that the state of any residue is 'felt' energetically by the other two upon mutation.
4.6 Pairwise coupling
241
Table 4.18. Free energies of pairwise coupling (in kcal/mol) for the stability of the transition state of subtilisin (Carter and Wells, 1988). Pair Asp32-His64 Asp32-Ser221 His64-Ser221
0
1
-7.88 ± 0.06 -6.59 ± 0.06 -8.94 ± 0.06
-0.11 ± 0.06 1.18 ± 0.06 -1.17 ± 0.06
D32H64S221 0.00+0.03 AHS 6.52±0.03
DAS 8.84±0.03
DHA 8.93±0.03
AAS 7.48±0.03
AHA 8.86+0.03
DAA 8.83+0.03
AAA 8.65+0.03
Figure 4.26 Schematic representation of the complete set of mutants of the catalytic triad Asp32, His64 and Ser221 of subtilisin (Carter and Wells, 1988). The free energy in the transition state (in kcal/mol) relative to the wild-type, as derived from eq (4.164), is shown for each mutant. The pairwise coupling patterns derived from the set of mutations are given in Table 4.18.
A similar effect is observed in mutational studies of subtilisin, where the spatially adjacent residues of the catalytic triad, Asp32, His64 and Ser221, have been mutated to Ala in a complete set (Carter and Wells, 1988). In this case, the free energy measured in the thermodynamic cycle in Figure 4.23 is the difference in the transition state free energy between the mutant and the wild-type, i.e., AAG| = m u t AG| - w t AG| (4.164) and the results are given in Figure 4.26. A positive value of this parameter indicates that the mutation has reduced the specificity of the enzyme and vice versa for negative values. The pairwise coupling pattern for each of the three possible pairs is given in Table 4.18. A drastic difference is observed upon mutation of the third residue, thereby indicating that the members of the catalytic triad act as a 'concerted switch' where each residue is influenced energetically by the others.
242
4 Specific cases and applications
The criterion of spatial proximity linked to indirect coupling of the concerted type, as envisioned by the picture dealt with in eqs (4.156)(4.161), is, however, not valid in general. We have shown in Section 4.5 that the protonation of the spatially close ionizable groups of glutamic acid can be explained fully in terms of direct coupling among the groups. A similar coupling mechanism is also seen for the four hemes of hemoglobin that are quite far apart from each other (see Section 4.4). Mutational studies on glutathione reductase have been carried out to perturb Alal79, Argl98 and Arg204, three residues not in close contact (Scrutton et al., 1990). The set of mutations is given in Figure 4.27, along with the relevant free energy values for the perturbation of the transition state in the oxidation of glutathione by NADH and NADPH. The pairwise coupling pattern given in Table 4.19 shows that the three residues are linked indirectly, although they are not in spatial proximity. The rules for communication in a biological macromolecule may be quite subtle. More information on higher-order perturbations is to be gathered and analyzed in a model-independent way using pairwise coupling patterns. The combination of massive mutational analysis of proteins and the principles of site-specific thermodynamics will eventually unravel general rules for gaining a predictive understanding of protein function and cooperative transitions from structure. Another application of the approach based on pairwise coupling can be developed in the case of studies on molecular recognition. In this case, the goal is to identify regions that are important for specificity and the
A179R198R204 0.00,0.00 GRR -1.10,0.08
AMR -0.62,2.68
ARL 0.41,2.42
GMR -1.32,2.11
GRL -1.54,0.87
AML -0.51,3.70
GML -1.72,2.22
Figure 4.27 Schematic representation of the complete set of mutants of residues Alal79, Argl98 and Arg204 of glutathione reductase (Scrutton et al, 1990). The free energy in the transition state (in kcal/mol) relative to the wild-type, as derived from eq (4.164), is shown for each mutant. The two values refer to oxidation in the presence of NADH (left) or NADPH (right). The pairwise coupling patterns derived from the set of mutations are given in Table 4.19.
4.6 Pairwise coupling
243
Table 4.19. Free energies of pairwise coupling (in kcal/mol) for the stability of the transition state of glutathione reductase (Scrutton et al., 1990). Pair NADH Alal79-Argl98 Alal79-Arg204 Argl98-Arg204 NADPH Alal79-Argl98 Alal79-Arg204 Argl98-Arg204
0
1
0.40 ± 0.06 -0.85 ± 0.06 -0.30 ± 0.06
0.74 ± 0.06 -0.51 ± 0.06 0.04 ± 0.06
-0.65 ± 0.06 -1.63 ± 0.06 -1.40 ± 0.06
0.07 ± 0.06 -0.91 ± 0.06 -0.68 ± 0.06
coupling among the various structural elements. The analysis can be carried out on either the 'macromolecule' or the 'ligand'. First, the subsites involved in the recognition mechanism need to be identified by single site-directed mutations. The quantity of interest can be the binding affinity of the ligand or the kC2Lt/Km ratio for the substrate. The perturbation of these signals due to mutation is used as a measure of the appropriate AG in the thermodynamic cycle of Figure 4.23. Second, a set of mutants is constructed to map the entire specificity domain and to derive the pairwise coupling pattern for the system. Assume that the system of interest is the specificity pocket of thrombin, a serine protease involved in blood coagulation. This pocket entails three subsites, as shown in Figure 4.28, and its structural organization sets the preference for substrates with Arg at PI, Pro at P2 and Phe at P3 (Bode, Turk and Kanshikov, 1992). The coupling mechanism among these subsites can be explored with mutants of thrombin, or using synthetic substrates that carry mutations at PI, P2 and P3. The latter alternative is clearly more convenient in practice. The set of synthetic substrates can be cast in vectorial form -S3-S2-S1-S1'-S2'-S3'-
-X-X-X-P3-P2-P1 -P1 '-P2'-P3 '-X-X-XFigure 4.28 Active-site mapping of a serine protease, using the strategy based on pairwise coupling. The specificity subsites of the enzyme SI, S2 and S3 (Schechter and Berger, 1967), interact with the residues PI, P2 and P3 of the substrate. Similar interactions involve the primed subsites. A complete set of mutants of either the S or P subsites yields the precise mechanism of coupling in the transition state.
244
4 Specific cases and applications
as follows [000] [100]
[010]
[001]
[no] [ioi] [on] [111]
(4165)
and is analogous to the ligand binding case dealt with in eq (4.39), and the examples given in Figures 4.25-4.27. The wild-type sequence FPR is perturbed stepwise to generate a triple mutant VAK, where Phe at P3 is replaced by Val, Pro at P2 is replaced by Ala, and Arg at PI is replaced by Lys. Hence, XXFPRXX XXVPRXX XXFARXX XXFPKXX XXVARXX XXVPKXX XXFAKXX (4.166) XXVAKXX is the set of substrates that 'chemically' maps the manifold in eq (4.165). X refers to an arbitrary residue. Once the ratio kcat/Km is determined for each substrate, the pairwise coupling free energy between any two subsites is derived from the transition state free energies of the four species in the thermodynamic cycle in Figure 4.23. For example, the quantity AGC = -RTln
( (^ gives the coupling free energy in the transition state between PI and P3, when there is no mutation at P2. The pairwise coupling pattern reveals how the three residues of the substrate are coupled in the transition state. This provides direct information on how thrombin subsites recognize these residues and are themselves coupled. A similar strategy can be used with protein-protein, protein-DNA or protein-RNA interactions. In the case of enzymatic cleavage, the quantity of interest is the transition state free energy. In the case of equilibrium binding, the signal is provided by the binding free energy. A combined approach with mutations involving the macromolecule and ligands probing different domains can yield all the information necessary to break the code for the communication among structural components. The higher the dimensionality of the functional space generated by the mutations with triple, quadruple and higher-order substitutions, the more complete the picture we obtain of macromolecular interactions using the approach based on pairwise coupling.
5 Site-specific effects in Ising networks
In this chapter we deal with the thermodynamics of site-specific effects in Ising networks (Newell and Montroll, 1953; Hill, 1960; Stanley, 1971; Domb and Green, 1972; Itzykson and Drouffe 1989) of particular biological relevance. The basic results of the phenomenological theory described in the foregoing chapters can be merged with those of the statistical thermodynamics of model systems that have set conceptual landmarks in our understanding of cooperative phenomena. A number of important properties of biological macromolecules have been approached in terms of lattice statistics, e.g., helix-coil transitions (Zimm and Bragg, 1959; Glauber, 1963; Cohen and Penrose, 1970; Poland and Scheraga 1970), protein folding (Dill et ai, 1995) and ligand-induced conformational transitions (Herzfeld and Stanley, 1974; Hermans and Premilat, 1975). These processes entail cooperative transitions involving a number of subsystems. Molecular recognition, or the process that allows proteins to interact specifically with small ligands, other proteins, RNA and DNA molecules, can be modeled in terms of the contribution of several individual recognition subsites that give rise to the macroscopic energetics measured experimentally. The properties of subsites can be modeled using Ising networks (Keating and Di Cera, 1993). A mechanistic description of cooperative processes should capture essential features which can be used eventually to develop more elaborate and accurate descriptions. An exact treatment of cooperative transitions in systems composed of a large number of interacting components remains, at present, computationally unfeasible. Interest in the analysis of site-specific effects in Ising networks also arises from the fact that the site-specific description can shed new light on outstanding problems related to properties of these networks in two, three and higher dimensions. Furthermore, these networks can be used to model site-specific effects under nonequilibrium conditions, such 245
246
5 Site-specific effects in Ising networks
as those pertaining to the hydrogen exchange properties of peptides and proteins (Englander and Kallenbach, 1984; Bai, Milne, Mayne and Englander, 1994; Zhou et al., 1994). It seems, therefore, quite appropriate to conclude the treatment of site-specific effects with a mechanistic description that complements and enriches the phenomenological analysis dealt with in previous chapters. Much of our understanding of site-specific thermodynamics of extended systems, along with their nonequilibrium properties, will depend on our ability to exploit the results of the phenomenological theory in the analysis of the Ising problem. 5.1 Transition modes
An Ising network consists of a set of N units connected through a given geometry in arbitrary dimensions (Newell and Montroll, 1953; Domb and Green, 1972; Itzykson and Drouffe, 1989; Keating and Di Cera, 1993). Each unit exists in two distinct states, which can be denoted by 0 and 1 without loss of generality. The network is called homogeneous if the units are equivalent and heterogeneous for all other cases encompassing differences among the units or asymmetric interaction geometries. The ring shown in Figure 5.1 is an example of a homogeneous network where all units are identical and equivalent since they contact two other units. The linear chain shown in Figure 5.1 is an example of a heterogeneous network. Although its units are identical, the end units differ insofar as they contact only one other unit. Heterogeneous networks in two or three dimensions are generated by randomly connecting an arbitrary set of identical or different units. If two units contact each other in the network,
Figure 5.1 Examples of Ising networks. The ring (left) is a homogeneous network since all units are identical and interact equally. The linear chain (right) is heterogeneous. Although its units are identical, they do not interact equally.
5.1 Transition modes
247
there is a nearest-neighbor interaction which depends only on the states of the units. Two ways of defining the rules for interaction have historically received much attention. The spin-lattice type of interaction assigns an interaction energy to two neighbor units in unlike states (Onsager, 1944; Lee and Yang, 1952; Newell and Montroll, 1953). The lattice-gas type of interaction assigns an interaction energy to two neighbor units in state 1 (Mayer and Mayer, 1940; Yang and Lee, 1952). Each treatment has advantages and fundamental properties. The spin-lattice rule encapsulates some general symmetry properties, while the lattice-gas rule simplifies considerably the mathematics when it comes to dealing with site-specific effects. A rather basic quantity in an Ising network is the energy spent per unit in the 0—> 1 transition. Intuitively, if a given unit is decoupled from the rest of the network, then the energy spent in the 0-» 1 transition for this unit is equal to the intrinsic energy for switching from state 0 to state 1. However, in the more interesting and realistic case where the unit is coupled to the rest of the network, the energy spent in the 0 —» 1 transition will also depend on the state of all other units. This conclusion is reached independently of the nature of the two states, 0 and 1, and the driving force responsible for the 0—> 1 transition. If the two states represent two different conformational states of individual domains of a macromolecular system at constant P, and the temperature of the heat bath provides the driving force for the 0—> 1 transition, then the network can be treated as a mechano-thermal ensemble. Alternatively, the 0—> 1 transition can model the binding of a ligand to a specific site of the macromolecule, at constant P and T, with states 0 and 1 denoting the free and bound forms. In this case the driving force for the transition is the chemical potential of the ligand and the network can be treated as a generalized ensemble. In the following, we will refer to states 0 and 1 as the free and bound forms of a site, modeled as a unit in the network. We assume T and P constant and use the generalized ensemble. We know from site-specific thermodynamics that for a system of N sites existing in two states there are 2N possible configurations, each of which is associated with a N-dimensional vector [d\ = [ o 1 d 2 . . . o j . . . e N ]
(5.1)
where 07 = 0, 1 labels the state of the yth site in the network. Each configuration is characterized by a Gibbs free energy specified by the states 0 and 1, and the interaction network among the sites. If the configuration with all sites in state 0 is taken as reference, then a total of 2N — 1 independent free energy terms characterize the network,
248
5 Site-specific effects in Ising networks
each corresponding to the work spent in the transition from the reference configuration to any one of the remaining configurations. These terms completely define the partition function of the system in the site-specific scenario. The energetics of the network can also be cast in terms of the free energy spent per site in the 0 —> 1 (free —> bound) transition. It follows from the phenomenological theory of global effects dealt with in Section 2.3 that the mean ligand activity is related to the average work spent in ligating one site of the macromolecule. This quantity refers to the work spent in the transition from the fully-unligated to the fully-ligated form, divided by the number of sites. However, the average work spent in ligating one site, or in the 0—>1 transition, can be defined from many other transitions. This point can be illustrated with the Ising networks depicted in Figure 5.2 in the form of graphs (Gould, 1988). The possible configurations are denoted by vertices and labeled accordingly. For TV = 1, the free energy spent per site in the 0 -»1 transition coincides with the mean free energy of ligation. For TV = 2, on the other hand, the free 01
000
Figure 5.2 Graphs representing the transitions of Ising networks of 1, 2 or 3 sites. Each possible configuration is labeled by a binary code and is depicted as a vertex in the graph. The connection between two vertices indicates the existence of a one-step transition that generates one configuration from the other by switching the state of a single site.
5.1 Transition modes
249
energy of transition can be defined in a number of ways. The transitions 00-^ 01 and 00 —> 10 provide two possible values. The transitions 10—> 11 and 01—> 11 provide two other values. The transition 00—>11 defines a free energy change for switching two sites to state 1 and when this value is divided by two, it gives yet another measure of the mean free energy change spent per site in the 0—> 1 transition. This particular value is the mean free energy of ligation defined in the global description. The five possible values listed above need not be the same, unless the sites are identical and independent. In the case of N = 3, there are transitions like 000-> 001, 001-> 011 and 011-> 111 that involve one site, those like 000—> 011 and 001 —> 111 that involve two sites and the transition 000—> 111 which involves all sites. For arbitrary N, there are transitions involving j = 1,2, . . . N sites, each providing in principle a different value for the mean free energy change spent per site in the 0—> 1 transition. Given any two configurations in the network, [a] and [o']9 the number of digits that one needs to change in [af] to obtain [o] is the 'Hamming distance', Haa>, between [a] and [af] (Hamming, 1986). Hence, the mean free energy spent per site in the 0—»1 transition, AG m (a, a'), can be written as AGm(
(5.2)
oo
Given any two configurations in the network, [a] and [a'], such that [a'] can be obtained from [a] by switching n sites from 0 to 1, the value of AG m (a, o') is the free energy change for the [a] -» [a f] transition divided by ft, i.e., the Hamming distance Hoo> between [o] and [a']. The transition free energy mode, or simply the transition mode (Keating and Di Cera, 1993) as defined in eq (5.2), is a sort of local 'intensive' property of the system which labels the energetics in a site-specific manner. A pictorial representation of transition modes is given in Figure 5.3 for the case N = 3. The configurations [a] and [o'\ are depicted by filled circles and the Hamming distance is the number of edges connecting them. Each edge is a valid transition step provided it increases the number of sites in state 1 by one. The transition mode is the total free energy spent in the trajectory, divided by the number of steps. When all possible transitions between allowable pairs of [a] and [af] are considered, a distribution of transition mode values is obtained. This distribution reflects the energetic landscape of the network, a sort of spectral decomposition of the free energy values for the 0—>1 transition. Essentially, a transition mode is the mean free energy of 0 —> 1 switching when the state of up to
250
5 Site-specific effects in Ising networks
4=^4
ytA=J[
N — 1 sites is held fixed. The operation of fixing M,0^ M ^ N — 1, sites in a given state has already been dealt with in the phenomenological theory in Section 3.2 and is called a contraction. The set of sites whose state is fixed is the contraction domain and the set of unconstrained sites is the free domain. Among the M sites in the contraction domain, a number /, 0 ^ / ^ M, may be fixed in state 1. The total number of distinct contractions, and hence transition modes, that can be generated in this way is v N = 3 N - 2N
(5.3)
This is because the number of ways of selecting M contracted sites is CM,N = N\/(N — M)\M\, and likewise the number of ways of selecting j sites in state 1 is C7?M = M\j{M — /)!/!. Hence the value of v^ is given by the double sum over M and j
5.1 Transition modes N-l V
N = E
251
M
1CM,NCJ,M
= 3" -2N
(5.4)
M=0 ;=0
Alternatively one can observe that each of the N sites has three possible states: contracted in state 0, contracted in state 1 and free, giving 3^ possibilities in all. Of these, 2N where all TV units are contracted in either state 0 or 1 must be eliminated. Hence, the transition modes are merely the mean free energies of ligation derived from the ensemble of contracted partition functions of the system. This is easily understood from Figure 5.2. In the case N = 1 there is only one contracted partition function of order zero and hence only one possible mean free energy of ligation. In the case N = 2 there is one zero-order contracted partition function and four first-order contracted partition functions, giving a total of five possible values of the mean free energy of ligation per site. In the case N = 3 there are one zero-order, six first-order and twelve second-order contracted partition functions, giving a total of nineteen possible values of the mean free energy of ligation per site. Contracted partition functions of the same order as the number of sites are not included in eq (5.4), since they are all equal to one and do not depend on the ligand activity (see Section 3.2). As to the free domain, we note that a transition mode can obviously be defined for any subsystem where the number of free sites is N — M. Consider an arbitrary configuration with M sites in the contraction domain, j of which are in state 1, while the remaining N — M sites in the free domain are all in state 0. Let [r] be the label for the vector associated with the configuration of the contracted sites. Then, MAGm(iV, Af,/) is the transition mode associated with [r] and represents the free energy change for switching all sites in the free domain from state 0 to 1, divided by N — M. For a fixed value of M and /, there are as many as CMyNCjiM possible values of ^AGm(N, M, j). We are interested in the distribution of AGm(N, M, j) values in the space of the vN possible contractions on the network and in how many of the contractions contribute to each transition mode. In principle, for a system where all sites are different and interact differently, such as the one dealt with in the phenomenological description, as many as v^ different values can be expected. The transition modes for CO binding to human hemoglobin are shown in Figure 5.4. The number of modes is very close to the actual number of contracted partition functions listed in Table 4.6. In the case of Ising networks, a high degeneracy of the transition modes is to be expected due to symmetry. In a homogeneous network where all sites are equivalent, each
5 Site-specific effects in Ising networks
252
AGm (kcal/mol) -10.0
-9.0
-8.0
-7.0
-6.0
-5.0
Figure 5.4 Transition modes for CO binding to human hemoglobin, obtained from the mean ligand activities associated with the contracted partition functions listed in Table 4.6. The values for the site-specific binding constants are listed in Table 4.8 and were derived from analysis of CO intermediates (Perrella, 1995).
transition mode has a degeneracy of w(N9 M, j) = CMtNCjiM. The unligated state 0 of a site is used as reference in the free energy scale, and AGS is the free energy of 0—»1 switching of an isolated site. Two sites contact one another when in the graph associated with the network they are connected through a single edge. Specifically, any given site in the ring shown in Figure 5.1 contacts two sites, while the end sites in the linear chain contact only one site. If a site contacts r others in the network, its connectivity is r. In a homogeneous Ising network, r is uniform everywhere and is a measure of the connectivity of the network as a whole. In a heterogeneous network the connectivity can only be defined as an average, since different sites may contact a different number of other sites. We now prove a theorem on the distribution of transition modes that holds for any Ising network. We choose the spin-lattice interaction rule, so that two sites in unlike states contribute an additional interaction free energy U to the total. The sign of U defines the nature of cooperativity, positive (U > 0) or negative (U < 0). Consider an arbitrary configuration of the sites in the contraction domain, with all sites in the free domain in state 0. The free energy associated with this configuration of N sites, relative to the reference with all sites unligated, is defined by the number of sites in state 1 and the number of 01 contact pairs in the network. Since each site in state 1 contributes AGS to the free energy, and each 01 pair contributes t/, a given configuration with M units in the contraction domain and N — M units in state 0 in the free domain is characterized by a free energy
5.1 Transition modes
AG0(N, M, ;) = ;AGS +
[XQ(N,M,
j) + v W M9 j)]U
253
(5.5)
where Xo(N, M, /) is the number of 01 pairs across the boundary between the free and contraction domains, and the suffix zero labels the state of all sites in the free domain. The coefficient ip(N', M, j) gives the number of 01 pairs inside the contraction domain. Upon switching all sites in the free domain to state 1 one has N9 M, j) = (N — M + ;)AGS + [xi(N, M, /) + 1/>(JV, M, (5.6) where ;fr(Af, ^> /') denotes the number of 01 pairs across the boundary between the free and contraction domains when all sites in the free domain are in state 1. The transition mode associated with the foregoing transformation is AGm(N, M, ;) = AGS +
[/ (5.7) N - M and is independent of \p{N', M, /). In the special case where M = 0, there is no contraction domain and eq (5.7) gives the mean free energy change per unit in switching from the reference configuration to that where all sites are in state 1. We know from the phenomenological theory that this transition mode is related to the mean ligand activity, xm9 and we will refer to it henceforth as the standard transition mode AG^ = AGS. The sum of the possible 01 pairs Xi(JV> M, j) + Xo(N, M, /) is the total number, ;&ot(JV> M), of physical contacts between the domains. Hence (Keating and Di Cera, 1993), Theorem: The distribution of transition modes for an Ising network is the sum of sub distributions symmetric about the standard transition mode AG°m and itself possesses this symmetry. To prove this theorem let us introduce the dual domain, defined as that obtained from the contraction domain by switching the state of all the contracted sites. All those in state 1 switch to state 0 and vice versa. This leaves M unchanged and results in the replacement /—> M — /. The sum of the possible 01 pairs Xo and Xi for a given contraction domain and its dual is the total number of interdomain contacts, i.e., 9 Xl(N,
M, /) + ;fo(W, M, M - ;) = ***(#, M)
(5.8a)
M, j) + Xi(N, M,M-
(5.8b)
j) = Xm(N9 M)
254
5 Site-specific effects in Ising networks
The sum of the transition modes for any contraction domain and its dual is AGm(N, M, j) + AGm(N, M,Mj) = 2AG°m + Xi(N, M, j) + Xi(N, M,M/) - Xo(N, M, j) - Xo(N, M, M - ; ) ^ N - M (5.9) and from eqs (5.8a) and (5.8b) it follows that j) = 2AG°m (5.10) AGm(W, M, ;) + AGm(W, M,MHence, the transition modes of a given contraction domain and its dual are equidistant from the standard transition mode. Furthermore, since w(N, M, ;) = CMiNCUM = CMtNCM-jM = w(N, M, M - /), the intrinsic weighting of any contraction, and the degeneracy of the transition mode associated to it, is equal to that of its dual. For M fixed, the components in the distribution of transition modes generated by changing j occur in pairs symmetrically disposed about the standard transition mode AG^. The self-dual contraction, M = 0, also obeys this rule as a special case. Thus, the subdistribution for any given M is symmetric about AG^. The overall distribution of transition modes, which is the sum of these subdistributions, must also possess this symmetry. Hence the theorem. It should be mentioned that if the lattice-gas rule of interaction is employed, then a weaker form of the foregoing theorem holds, requiring that the network be homogeneous. The generality of the theorem in the case of spin-lattice interaction rule is quite interesting and bears on the results embodied by the Lee-Yang theorem of statistical mechanics (Lee and Yang, 1952). This theorem asserts that the partition function of an Ising network under the spin-lattice interaction rule is always symmetric, with the roots distributed on the unit circle in complex plane. The theorem on the transition mode distribution provides further information on the properties of the partition function by stating that all of its contracted forms are also symmetric. This corollary to the theorem follows from the fact that if a given configuration enters the definition of a contracted partition function, so does its dual with identical degeneracy. As for the Lee-Yang theorem, our results are independent of the dimensional embedding of the network, since the analysis is based entirely on combinatorial arguments. The connectivity of the network, r, and hence its dimensionality, comes into play when the range of each subdistribution for fixed M is to be determined. This quantity gives a measure of how the transition modes spread over the AG axis for a given value of M. Any given transition mode can be written as
5.2 Transition modes &Gm(N, M, j) = 2AG°m - AGm(N, M,M=
255 j)
AG[,_tt.(*,M)-2a(iV,*,J)
N - M The number of 01 pairs, Xi{N, M, j) varies from 0 for /' = M, to Xtoti^f M) for ; = 0. Hence, the limits for the transition modes for a given value of M are +
> M) U (5.12) N - M and we seek the maximum value of Xtot(N> M) for a given M. Suppose that M ^ N/r. Then there is a disposition of the contracted sites such that there is no contact within the contraction domain, in which case Xtot(Af, M) = rM and ±AGm(N, M, j) = AG°m ± M(N - M)~hU. Likewise, for M ^ N/r there is a disposition of the contracted sites such that there is no contact within the free domain, so that X\x>t(N, M) = r(N — M) and ±AGm(N, M, /) = AG°m ± rU. The transition modes of the overall distribution satisfy the condition -AGm(N, M, j) = AG°m ±
AG°m -rU^
Xtot(N
AGm(N, M, j) ^ AG°m + rU
(5.13)
Since the term ;£tot(N, M) — 2xi(N9 M, j) in eq (5.11) must be an integer, each subdistribution for M fixed has a characteristic spacing of U/(N - M). This is the minimum possible distance between two transition modes within the subdistribution. The set of values that Xtot(N, M) - 2xi(N, M, j) can take on depends on the details of the network, such as the value of r and the dimensional embedding. To illustrate the properties of the transition mode distribution we consider the special case of an Ising network where each site contacts every other site. In this case the connectivity is r = N — 1 and both Xtot(N,M) and ^(iV, M,y) are independent of the disposition of the contracted sites and are uniquely determined once N, M and / are specified. For any given contraction one has / to t(^> M) = M(N — M) and Xi(N, M, ;) = (M - j)(N - M), so that AGm(N, M, /) = AG°m + (M- 2j)U = AG°m + nU
(5.14)
where n = M - 2/ goes from -(N - 1) to (N - 1), since M = 0, 1, . . . N - 1 and j = 0, 1, . . . M. The transition modes for this network are uniformly distributed in the range ±{N - 1)C/, with a spacing of U. For fixed M, the distribution is symmetric around / = Af/2, as implied by the theorem, has a spacing of 2U and follows the binomial CjM. For N large, M is also large and the binomial distribution CjM goes into a Gaussian
256
5 Site-specific effects in Ising networks
with mean Af/2 and variance M/4, as implied by the De Moivre-Laplace theorem (Wilson, 1911; Feller, 1950). The Gaussian approximation is remarkably good even for small values of N. Hence, each subdistribution for fixed M can be approximated by a Gaussian distribution with mean and variance given by [i(N, M) = AG^ and o2(N, M) = MU2. The mean is independent of M, while the variance grows linearly with the size of the contraction domain, independent of N. This is equivalent to a Brownian motion in one dimension around the origin x = AG^, where M plays the role of time and the diffusion coefficient D is proportional to U2 (Chandrasekhar, 1943). In the absence of interactions, as expected for a reference system, U = 0 and no diffusion of modes is observed around the standard transition mode. The overall distribution is a 6-function peaked at the origin. For finite £/, and regardless of whether interactions stabilize (U < 0) or destabilize (U > 0) 01 pairs in contact, the mode distribution spreads out with increasing M under the effect of a 'drift' provided by the interaction energy U. The further away two configurations [a] and [af] in the ensemble are, the more likely that the transition mode would be close to the standard transition mode AG^. The greater the Hamming distance between two configurations, the smaller is the variance of the distribution of transition modes associated with it. Each subdistribution has a variance of MU2 and an intrinsic weight of 2M. The overall distribution of transition modes is the sum of all subdistributions obtained for fixed M9 times the number of ways of fixing M units among N9 CM^N. Some overlap exists among the subdistributions since all values of M and j such that M — 2] = n = constant contribute to the same mode. It is quite instructive to derive analytically the degeneracy, P(n), of a given mode as follows. By definition N-l
p
M
(n) = £ I,CMtNChMdmn
(5.15)
M=0 y=0
where the Kronecker delta, <5mn, filters out all values of m = M — 2] not equal to the prescribed value of n. The form of P{n) can be found by introducing the integral representation of 8mn (Pearson, 1990) 8mn
=
1
r2lT
2TT
Jo
exp [i(n - m)cp\ dcp
(5.16)
Interchanging eq (5.16) with the double sum in eq (5.15) yields 1
P(n) =
2TT
N l
~
M
f exp(in(p)2 ^CNMCMJQxp[i(2j 2TT JO
M=O
;=o
- M)cp]d(p (5.17)
5.1 Transition modes
257
Elementary transformations lead to P(ji) = _ L f^exp (\ncp)[{\ + 2cos
(5.18)
and symmetry of the integrand gives the equivalent form P(n) = - 1 pexp (m
(5.19)
In the limit of large N, the expression in square brackets has a dominant peak with maximum 3" at cp = 0. The value of the maximum is the expected value of vN in the same limit. The Taylor expansion around this point yields ~3 N exp - i l ^ l (5.20) (l + 2cos 8.0. Three main clusters of modes are clearly distinguishable at pH < 8.0 and tend to merge at higher pH. Third, in no case is the pattern consistent with that of
5 Site-specific effects in Ising networks
258 o
O
q d
o CO o
o
8
-20.0
-10.0
0.0
10.0
20.0
M-2j Figure 5.5 Transition mode distribution for a homogeneous Ising network with N = 20 and r = N — 1. The continuous curve was drawn according to eq (5.21), while points were calculated by computer analysis of the network exhausting all possible configurations (Keating and Di Cera, 1993). an Ising network with all sites coupled to the same extent, thereby suggesting that coupling in the hemoglobin molecule involves interactions of different strength that change drastically at pH > 8.0. These conclusions concur with the results of our analysis in Section 4.4. 5.2 A combinatorial model for molecular recognition The concept of transition mode can be exploited in the analysis of molecular recognition, a basic problem in physical and computational biochemistry that is tantamount to predicting free energies of interaction from structural components. Here we discuss a simple model that reduces molecular recognition to a combinatorial problem (Keating and Di Cera,
5.2 A combinatorial model for molecular recognition
6.7
7.2
7.7
8.2
8.7
9.2
259
9.7
Figure 5.6 Spectral representation of the transition modes for the Fell/FelllCN hemoglobin system as a function of pH. The modes were obtained from the mean ligand activities associated with the contracted partition functions listed in Table 4.6, using values for the site-specific binding constants derived by Daughetal (1994).
1993). Let us consider two surfaces, A and B, with A containing TV acceptor subsites for TV donor subsites of B as shown in Figure 5.7. For the sake of simplicity surface A is assumed to be inert, i.e., its energetic state is the same when free or bound to B. The donors of B can exist in two energetically equivalent states, 0 and 1. When surface B is free, all donors are in state 0. When surface B is bound to surface A all donors switch to state 1 in a concerted fashion. Each pair 'donor in state 1: acceptor' provides a free energy of stabilization, AG°, across the A-B interface of the bound complex. State 0 can be interpreted as a neutral state with the pair 'donor in state 0: acceptor' providing neutral stabilization to the bound complex. The free energy of binding of surface B to A is simply AG0 = NAG°. This is a very simple model for the interaction of
260
5 Site-specific effects in Ising networks
Figure 5.7 Schematic representation of molecular recognition as a combinatorial problem. Two surfaces, A (left) and B (right), form a complex involving the interaction of a number of complementary subsites. Each ensemble of subsites can be modeled as a two-dimensional Ising network and the free energy distribution for the binding reaction can be derived from the properties of the transition modes of the networks.
two macromolecules and can also be used in the analysis of the protein folding problem by allowing the two surfaces A and B to be distinct domains of the same macromolecule. Protein folding is, in fact, a problem of molecular self-recognition, as illustrated by the elegant work of Rose and his colleagues (Presta and Rose, 1988; Harper and Rose, 1993; Lattman and Rose, 1993; Aurora, Srinivasan and Rose, 1994). Assume now that a number of site-specific 'mutations' are made on surface B. We assume that any mutation may have the effect of 'freezing' one or more donors in either one of the allowable energy states, 0 or 1. In principle, there is no way one can predict the effect of the mutation on the free energy value for the formation of the AB complex, since the mutation can act locally at the subsite where it applies and/or propagate to distant sites. However, regardless of the extent and precise nature of the perturbation, we know that surface B free can assume only one of the 2N possible configurations allowed for the manifold of donors. Assume that the mutation freezes M donors, j of which are in state 1, and that the
5.2 A combinatorial model for molecular recognition
261
probabilities of freezing a donor in state 0 or state 1 are the same. In mathematical terms, the effect of the mutation is equivalent to generating a contraction domain in the system. The free energy of binding of this mutant to surface A can be computed as follows. A total ofN — M donors are free to switch to state 1 when bound to surface A. These donors provide a free energy change (N — M)AG°. Among the frozen donors, M — j remain in state 0 even when bound to surface A. These donors provide no contribution to the binding free energy. On the other hand, the j donors frozen in state 1 provide an additional term /AG° to the total free energy of binding, which is then AG(7V, M, j) = (N - M + j)AG°
(5.22)
The binding free energy difference between the mutant and the wild-type surface B is evidently AAG(N, M, /) = - ( M - ;)AG°
(5.23)
In this simple model any mutation is expected to reduce the binding affinity, unless the number of donors frozen in state 1 coincides with the total number of frozen donors M. The entire spectrum of AAG values for the system is generated by changing M and j . For a fixed value of M, the values of AG(N,M,j) are distributed as a Gaussian with mean fj(N, M) = (N - M/2)AG° and variance o2(N, M) = (M/4)AG°2 . The probability that the binding free energy for any mutant is x is given by the expression
(i
"ii ( ,
) Up [ *[ - «"
M
>[|
(5.24)
where w(N, M) is an ad hoc weighting factor for the subdistribution of free energy values obtained for a given M. The basic assumption of the foregoing model is that the system (surface B) explores a unique 'free energy space' when structural perturbations are considered. Single- or multiple-site mutations can be analyzed using the same framework. Essentially, the number of allowable free energy values for binding is finite and is determined by the combinatorics of the network of donors. The predictions of this model can be tested by analyzing free energy profiles obtained for macromolecular systems and their mutant derivatives. The relevant parameters of the model can be obtained by analyzing the distribution of AG(N, M, /) values obtained experimentally in terms of a combination of Gaussian terms, whose mean and variance are to be considered as independent parameters. Each Gaussian term should be weighted properly using another set of independent parameters for the
262
5 Site-specific effects in Ising networks
filters w(N9 M). Once the best-fit values of \i and a2 are determined for all Gaussian terms, then a plot of a2 versus \i should be constructed. The model predicts that such a plot would be a straight line and specifically - y This plot yields the number of subsites TV and the free energy of binding provided by a single 'donor in state 1: acceptor' pair across the A-B interface. Both parameters are of considerable interest in relating perturbed energetics to structural features. In the case where interactions exist among the donor subsites of surface B, the treatment is more elaborate. We make use of the results derived in Section 5.1 and assume that each donor is in contact with r = N — 1 other donors. Two donors in contact interact according to an energy U only if they are in different states. The interaction network of this fully connected Ising network resembles a 'molecular field' (Hill, 1960; Itzykson and Drouffe, 1989), since the state of any given donor in the surface will be felt by all other donors in unlike states. The effect of a mutation on surface B is more complex than that discussed in the case of independent donors. When M units are frozen, / of which are in state 1, the energetic state of the surface can be predicted from the number of donors in states 0 and 1 and the interactions among them as GB(free) = j(N — j)U. When the surface is bound, all donors in the free domain switch to state 1, while the donors in the contracted domain remain frozen. The new energetic state of the surface is GB(bound) = (N - M + j)(M - j)U. The energetic state of the interface between A and B is computed by counting the number of 'donor in state 1: acceptor' pairs. The free energy of binding of the two surfaces is evidently AG(N, M, j) = (N - M + j)AG° + (N - M)(M - 2j)U (5.26) and reduces to eq (5.22) in the absence of interactions. The free energy of binding is closely related to the transition modes of surface B and the results derived in the previous section can be exploited in the analysis of the A-B interaction. It follows from the results in Section 5.1 that AG(N, M, /) is symmetric around AG(N, M, M/2) for fixed M and that the AG(N,M,j) values are uniformly distributed with a spacing of AG° — 2(N — M)U. The subdistribution for M fixed can be approximated by a Gaussian with mean and variance /
M \
li(N, M) = [N - —
AG°
(5.27a)
5.2 A combinatorial model for molecular recognition
263
, M) = — [AG° - 2(N - M)U] 2 (5.27b) 4 and the overall distribution of AG(N, M, j) values can be written as the superposition (5.24). The relationship between o2 and /J, is 2 (AG°2 + 2NUAG° 2 (5.28) a = (NAG° 2AG°3 The variance a2 is a cubic function of the mean \i, a very peculiar result. The linear dependence of a2 on \x in eq (5.25), observed in the case of independent donors, is recovered as expected for U = 0. The dependence of a2 on fi predicted by eq (5.28) has been tested and confirmed experimentally in the biologically relevant case of thrombinhirudin interaction (Keating and Di Cera, 1993). The distribution of oin CM
8 CN
d
o 6 o o
o ID o
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
AAG (kcal/mol) Figure 5.8 Distribution of binding free energies for thrombin-hirudin interaction relative to the wild-type (-17.93 ± 0.09 kcal/mol). The AG values obtained for sixty-seven mutants were grouped in bins of 0.4 kcal/mol to determine the number of mutants within each free energy interval. The curve was drawn according to eq (5.24) using five Gaussian components. The best-fit values of pi and a2 for each component are shown in Figure 5.9.
5 Site-specific effects in Ising networks
264
binding free energies is shown in Figure 5.8, as derived from the analysis of sixty-seven mutants of hirudin in their reaction with thrombin (Wallace, Dennis, Hofsteenge and Stone, 1989; Betz, Hofsteenge and Stone, 1991). These mutants have single- and multi-site substitutions of polar and apolar side chains over the entire surface of recognition, which spans about 12% of the available surface area of thrombin (Rydel, Tulinsky, Bode and Huber, 1991). The landscape of AAG values spans a range of nearly 10 kcal/mol, as shown in Figure 5.8, and is distinctly multimodal. Individual Gaussian components can easily be detected. Hirudin is modeled as surface B and thrombin as surface A. The continuous line depicts the best-fit of the experimental data to a superposition of Gaussian terms, whose mean and variance are given in Figure 5.9. The cubic dependence of o2 on \i predicted by the model is indeed seen experimentally. All independent parameters in eqs (5.27a) and (5.27b) can be resolved from nonlinear least squares. In particular, the number of recognition subsites, q CO
o
i
I 6 q d -20.00
-18.33
-16.67
-15.00
-13.33
-11.67
-10.00
ix (kcal/mol) 2
Figure 5.9 Dependence of a on \x for the five Gaussian components shown in Figure 5.8. The prediction drawn from eq (5.28) is confirmed experimentally.
5.3 Site-specific effects in Ising networks
265
TV, is found to be 12 ± 3, a value very close to the number of ion pairs and hydrogen bonds revealed by the crystal structure of the thrombin-hirudin complex (Rydel et ai, 1991). Each subsite contributes —1.5 ± 0.3 kcal/ mol to the binding free energy. The interaction energy U = — 0.10± 0.04 kcal/mol indicates that neighboring subsites are negatively linked and provides a measure of the energetic cost of switching one donor of the contraction domain to state 1. For example, if a mutation causes M = 1 and y = l, then the free energy cost that translates in a loss of binding free energy is — (N — 1)U = 1.1 kcal/mol. The example dealt with above should be considered paradigmatic for the development of more elaborate treatments of molecular recognition using Ising networks. It is quite interesting that a simple framework is capable of encapsulating the salient features of a protein-protein interaction and its perturbation by site-directed mutations. Clearly, the treatment used for surface B can be extended to surface A and both surfaces can be assumed to have their own combinatorial switching mechanism upon binding. Arbitrary networks of interaction among recognition subsites can be introduced, consistent with information about the detailed crystal structure of the complex. Also, different contacts can be assigned different energetics. In general, the two surfaces should be modeled as heterogeneous two-dimensional Ising networks and the combinatorics should be exhausted by computer for each particular case. Alternatively, an analytical treatment of such networks for systems of biological relevance should be pursued. 5.3 Site-specific effects in Ising networks
Homogeneous networks provide useful modes of cooperative transitions, but heterogeneous networks embody most of the fundamental features of biological macromolecules, as they allow for intrinsic differences among individual sites and asymmetric or arbitrary patterns of interactions. An important aspect of heterogeneous networks is that they lend themselves to the analysis of properties arising locally at the level of each individual site. This feature stands in contrast to homogeneous networks where symmetry renders the global properties of the system a scaled replica of those arising at any site in the network. Obviously, the mathematical analysis of heterogeneous networks is far more complex. There are essentially two ways of approaching the study of Ising networks. The classical approach is based on the 'transfer matrix' method and allows for calculation of the partition function of the network in many cases of
266
5 Site-specific effects in Ising networks
interest (Onsager, 1944; Newell and Montroll, 1953; Poland and Scheraga, 1970; Stanley, 1971). The other approach is somewhat more general and is based on consideration of important topological properties of the network. We will start our discussion with the former approach, which is more 'technical' and then consider aspects of the latter approach, which is more 'intuitive'. Consider the one-dimensional Ising ring in Figure 5.1. If all sites are equivalent, the ring is a homogeneous network for which global and local properties are derived by simple rescaling. We assume the lattice-gas rule of interaction (Mayer and Mayer, 1940; Yand and Lee, 1952; Hill, 1960) because it simplifies the mathematics significantly, and define the interaction constant between two neighboring sites in state 1 as a. Positive or negative coupling between neighboring sites correspond to o > 1 or a < 1 respectively. The reference system is obtained for a = 1. We start with the analysis of the ring to illustrate the basic properties of the transfer matrix. Consider two neighboring sites on the ring, as in Figure 5.10, and their four allowable configurations. The transfer matrix is an operator that encapsulates the allowable states as follows
W = l1
M
(5.29)
v 7 \(o acoj The meaning of each term of W can be understood as follows. If the two sites in Figure 5.10 are considered in terms of the reference cycle dealt with in Section 3.1, the partition functions of site j when site / + 1 is unligated or ligated are respectively
°Vy+1 (*) =l + Kjx = l + Kx = l + a)
(5.30a)
lx
(5.30b)
¥ j+1(x)
= 1 + oKjX = 1 + oKx = 1 + aco -O-O-
l
-•-O-
Kx
_O-#-
Kx
-•-•-
*K2x2
Figure 5.10 Possible configurations of a nearest-neighbor pair including sites / (left) and / + 1 (right) in the Ising ring shown in Figure 5.1. Open and filled circles depict free and bound states respectively. Contacts between sites are indicated by a connecting line. The statistical term weighting each configuration is also indicated, assuming the lattice-gas rule of interaction. K is the site-specific binding constant, x the ligand activity and a the interaction parameter.
5.3 Site-specific effects in Ising networks
267
The terms in eqs (5.30a) and (5.30b) involve the site-specific binding constant for site /, Kj, the interaction constant, a, and the ligand activity, x. Since the sites are identical, Kj = K for all sites and, for the sake of simplicity, the product Kx can be used to define a dimensionless activity variable w. The meaning of a) is that of a ligand activity measured in K~l units. The terms defining the contracted partition functions (5.30a) and (5.30b) are the same as those entering the definition of the transfer matrix W. The first column of W is constructed from the terms of °^/;+i and labels the configurations of site j with site j + 1 unligated. Likewise, the second column of W is constructed from the terms of 1V7-+1 and labels the configurations of site j with site j + 1 ligated. Hence, the elements of the transfer matrix W can be mapped into terms defining specific configurations in the contracted partition functions derived for each neighboring pair. The partition function of the ring is obtained by multiplying together the transfer matrices for all possible neighboring pairs around the ring and taking the trace of the product (Hill, 1960; Poland and Scheraga, 1970; Stanley, 1971), i.e., W(co) = Tr(W 12 W 23 . . . VTN-1NVfN1) (5.31) This is the general solution for a ring of N sites, with N distinct transfer matrices containing different interaction constants for each pair and different binding constants for each site. In the specific case of a homogeneous ring, the sites bind with the same affinity and interact equally, so that there are N equivalent pairs and eq (5.31) simplifies to W(co) = TrW" (5.32) The explicit form of W is obtained by switching to an orthogonal basis set. This is equivalent to diagonalizing W, a transformation that leaves the trace unchanged (Gantmacher, 1959). The similarity transformation is well known in matrix algebra and takes on the familiar form W((o) = TrW" = Tr A" = Af + A2" (5.33) where A is the diagonal form of W. The eigenvalues of W are the solutions of the secular equation . (O
GO) - A
= (1 - X)(aco - A) - a> = 0 V
/V
7
Hence, oca ± V ( l - o(o)2 + 4co
V
(5.34) y
5 Site-specific effects in Ising networks
268
Once *P is known, all response functions of interest are derived from differentiation as dealt with extensively in Chapter 2. The number of ligated sites is X = d In W/d In co and the binding capacity is B = dX/dln co. The function X is shown in Figure 5.11 for various values of the interaction parameter o. The case a = 1 yields the properties of a reference system as already dealt with in Chapter 2. We now consider a simple modification of the Ising ring that is of particular biological relevance and that leads to consideration of sitespecific effects. If the ring is 'perturbed' by eliminating the contact between two sites, say 1 and N, a linear chain is obtained as shown in Figure 5.1. Although the sites bind with the same affinity K, the perturbation introduces an asymmetry in the network. As a result, the sites are no longer equivalent and the network as a whole is heterogeneous. The o
CO
q
q
q d -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
log co Figure 5.11 Binding curves for the homogeneous Ising ring for N = 6. Curves were drawn using the partition function in eq (5.33) with values of a = 10 (left), a = 1 (middle) and o = 0.1 (right). The values of o correspond to positive, no and negative cooperativity respectively.
5.3 Site-specific effects in Ising networks
269
partition function in this case involves N - 1 transfer matrices of the same form as W, but also two end vectors such that (Poland and Scheraga, 1970) *P(a>) = L+W^-'L.
(5.36)
where L + = (1
1)
(5.37a)
L_ = p
(5.37b)
Calculation of the partition function is lengthy, but straightforward. However, instead of pursuing the matrix approach, which is somewhat tedious in the general case, we shall use a different approach to derive W for the linear chain. The advantage of this alternative approach, which exploits contracted partition functions, will soon be evident. We use the fact that in the lattice-gas rule the interaction constant o between neighboring sites only applies when both sites are in state 1. Hence, keeping a site in its unligated state is equivalent to deleting the site from the network. In fact, neighboring sites will not 'feel' the presence of the site being frozen in a particular state, unless it is in state 1. Freezing a site in state 0 among N sites leaves the properties of the remaining N — 1 sites unaltered. This fact has far-reaching consequences. Consider any of the N sites of a homogeneous ring. The site-specific binding curve of this site is derived from the phenomenological theory as = l- - 1 * ^ (5.38) W() where *P is the partition function of the ring. Since the network is homogeneous, it makes no difference which site is considered and Xt = 6 = X/N = (l/AOdlnW/dlnco for all sites. The contracted partition function in the numerator is obtained from that of the ring when site i is frozen in its unligated form. This is equivalent to dropping site i from the ring and, since all units are equivalent, to generating a linear chain of N — 1 sites from a ring of N sites. The contracted partition function in the numerator of eq (5.38) is therefore the partition function for a linear chain ofN — 1 sites. Rearranging eq (5.38) and using the foregoing definitions leads to Xi
L^«
N + 1 din sites Knowledge of the partition function of a ring of N + 1 sites,
(5.39) Rl Rv
PAr+1,
270
5 Site-specific effects in Ising networks
defines the partition function of a linear chain of N sites, Lx¥ N, through the transformation in eq (5.39). Using the eigenvalues of the transfer matrix and eq (5.33) yields ^WN(o>) = Af (Ai - A{) + A2N(A2 - A0
(5.40)
where A{>2 = dA1?2/dln wis given by «, _ oco\/(l — aco)2 + 4co ± o)[2 — o{\ — Ma 2V(1 - ocof + 4co
Consideration of contracted partition functions yields a straightforward derivation of the properties of the linear chain from those of the Ising ring. The foregoing method can be extended to the analysis of site-specific effects in the linear chain. Unlike the ring, the linear chain is heterogeneous and lends itself to consideration of site-specific effects. Quite interestingly, analysis of these effects can be parameterized in terms of the interaction constant a alone. The site-specific properties of the linear chain can be derived by consideration of the basic eq (5.38), which applies regardless of the particular model being considered. In the case of a linear chain, the contracted partition function 0}¥i has a very simple interpretation, as shown in Figure 5.12. When site i is kept in its unligated state, the chain is cut into two smaller chains containing i — 1 and TV — i sites. Hence, °V (5.42) f(fi)) = ^U^N-tico) and each term on the right-hand side of (5.42) is of the form (5.40). The partition function of the linear chain can be rewritten in a more compact form as follows = Ar + 1 [l - * ! + ( ! - a2)AN+1] = Af +1 (a 2 + ^ A ™ )
(5.43)
O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O-O 1 2
O-O-O-O-O-O-O-O
i
N
O-O-O-O-O-O-O-O
1 2 i-\ i + l i+2 N Figure 5.12 Derivation of the site-specific properties of a linear chain. From eq (5.38) it follows that the site-specific binding curve of site i is defined by the partition function of the chain containing iV sites (top) and its contracted form with site / kept unligated. This contraction generates two smaller chains of / — 1 and N — i sites (bottom).
5.3 Site-specific effects in Ising networks
111
using the definitions ocx = A{/A1? a2 = A2/A2, A = A2/A!. It is easy to verify that oc\ + a2 = 1 and — l ^ A ^ l . In particular, A is a parameter of smallness, since it gives the ratio between the minimum and the maximum eigenvalues of W. The site-specific binding isotherm written in terms of the foregoing definitions is given by . - A')(l - A"-' + 1 )
(5.44) a2 and is plotted in Figure 5.13 for N = 9. As expected on intuitive grounds, the sites in the middle of the chain behave more cooperatively than those toward the ends. Symmetry around the mid-point M = (N + l)/2 is also evident from eq (5.44). Sites equidistant from M have the same binding curve, since the replacement i ~> N - i + 1 leaves eq (5.44) unchanged. =
00
6
CO
d
q 6 -2.0
-1.5
-1.0
-0.5
0.0
log co Figure 5.13 Site-specific binding curves for the sites of a linear Ising chain for N = 9. Curves were drawn using eq (5.44) with values of a = 10 and refer to sites 1-5 from right to left. The curves relative to sites 6-10 are identical to those of sites 1-4 because of the symmetry of the chain around the site in the middle (site 5).
272
5 Site-specific effects in Ising networks
Particularly interesting is to express Xt relative to the value of XM, i.e., the site-specific binding curve of the site in the middle of the chain. For the sake of simplicity we assume that TV is odd. From eq (5.44) it follows that axa2\ 1 a2 + axAN+1
Hence, j — XM
(1 - AQ(1 -
A )
(5.46)
For large N eq (5.46) can be approximated by Xj = XM(1 - A'') = XM[1 - exp(ylnA)].
(5.47)
For fixed co, the difference XM — Xj decays exponentially from the end of the chain to the middle, as illustrated in Figure 5.14 for different values of the interaction parameter a. The case a = 1 gives XM — Xj = constant throughout, as expected. The case a > l , corresponding to positive nearest-neighbor interactions, follows the envelope of the exponential decay. The case o< 1, corresponding to negative nearest-neighbor interactions, is characterized by damped oscillations of the function X M - Xj, with the damping term given by the exponential decay in eq (5.47). A basic property of the decay is the 'relaxation length'
1=
"rr^
(5 48)
-
ln|A| which follows directly from eq (5.47). This parameter provides a measure of the communication between sites in the linear chain. The absolute sign in eq (5.48) is necessary, because A < 0 for a < 1. The relaxation length is independent of the length of the chain and is uniquely defined by the eigenvalues of the transfer matrix. The value of / as a function of co is plotted in Figure 5.15. The relaxation length is peaked at a point such that awmax = 1, as can be proved easily from the definition of A. Memory effects in the linear chain reach their maximum for this value of co. The value of / at the peak is /max = "
"
In
V7
(5.49)
- 1
5.3 Site-specific effects in Ising networks
273
CO
6
d
d
CO
d
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0 10.0 11.0
Site Figure 5.14 Relaxation properties of the linear chain shown as the difference XM - Xj versus / = 1, 2, . . . (N + l)/2, for a fixed value of co and AT = 19. The continuous lines depict the envelope of the exponential decay embodied by eq (5.47). Points refer to: (open circles) a = 10, co = 0.1; (filled circles) o = 1, co = 1; (stars) a = 0.1, co= 10.
A value of /max = 0 is obtained in the absence of interactions (o = 1), thereby denoting the absence of 'memory' along the chain. In this limit all sites behave identically. On the other hand, /max —> oo for cr—> 0, or a—> <*>, thereby indicating the presence of long-range correlation and communication among the sites of the chain in the case of strong coupling. Note that the replacement a—> a" 1 leaves /max unchanged. Values of 1.5 ^ /max ^ 5 are obtained for 10 ^ a ^ 100, which are typically seen in practical applications. The foregoing analysis allows us to tackle the rather interesting problem of how information arising from site-site coupling is communicated in an Ising network. Here we further exploit the simplicity of the lattice-gas type of interaction to understand how perturbations of a homogeneous Ising
5 Site-specific effects in Ising networks
274 o CO
q
q
CO
Pi
9 CN
o
q d -3.0
-2.0
-1.0
0.0
1.0
log co Figure 5.15 Relaxation length for the linear chain as a function of the logarithm of a). The value of / peaks at comax = a"1. The continuous and dashed lines refer to a = 10 and a = 100 respectively.
ring produce the heterogeneous linear chain. Consider the Ising ring in Figure 5.16. All neighboring pairs are coupled with an interaction constant a and have a site-specific binding constant K, with the exception of site 1, for which the binding constant is K'. The purpose of our analysis is to study the transition for a homogeneous network, obtained for Kf = K, to a heterogeneous linear chain, obtained for Kf = 0. Also, all cases obtained for K1 •=£ K deserve much consideration in a site-specific description, since they refer as well to heterogeneous networks. We shall refer to the Ising network in Figure 5.16 as the 'perturbed ring' and we seek to find the analytical expression for binding to site i =£ 1 in the ring. The basic eq (5.38) demands the definition of the partition function of the perturbed ring, W, along with its contracted form °lI//. Using the definition of contracted partition functions we have
5.3 Site-specific effects in Ising networks
275
Figure 5.16 Heterogeneous Ising ring. All sites interact equally, but site 1 (hatched circle) has a different site-specific binding constant, K'. In the limit of Kf —» 0, the ring turns into a linear chain oi N — 1 sites.
W((o) = "W^co) + 1W1(co)wf (5.50) f Ox where co = K'x. The partition function Pi is that of a linear chain of N — 1 identical sites, Lx¥ N-l9 whose solution is given by eq (5.40). The partition function lxU1 is the same as that of a homogeneous ring of N sites, with one site kept in its ligated state. For a homogeneous ring the partition function is the same as eq (5.50), except that co' should be substituted with co. Hence,
.„,,<„». »•"»(*> - L ^ - w
(5.51)
CO
and the partition function of the perturbed ring is given by (5.52) V(fl>) = v R ^ ( ^ ) + (1 - v^W^ico) f where v = a) /co= K'/K. As expected, W is the weighted sum of the partition functions of the homogeneous ring of N sites and the heterogeneous linear chain of N — 1 sites, with the ratio between K' and K being the weighting factor. The two limiting cases are obtained for v = 0 and v=l. The contracted partition function °VI- in the perturbed ring can be computed as follows. Contraction with respect to site / yields a linear chain containing TV - 1 sites, N - 2 of which bind with a site-specific constant K, and the perturbed site 1 binding with a constant K'. To obtain the partition function we further contract °Wh so that °^(co) = ^ ( c o ) +01Wn(co)(o' (5.53) The significance of the various terms is obvious. The partition function
276
5 Site-specific effects in Ising networks
mx
Vn is the product of two partition functions referring to linear chains of TV — i and i — 2 identical sites. The partition function Olx¥ n is the same as that of a linear chain of TV — 1 identical sites, with one site kept in the ligated state. For such a chain the partition function is the same as eq (5.53), with a)' substituted with co. Hence,
W^W
(5 54)
CD
The contracted partition function 0Wt is given by ^ ( r o ) = v^N-iico)
+ (1 - v) L ^_Kco) L W;- 2 (co)
(5.55)
l
As expected, °F / is the weighted sum of the partition functions of a linear chain of TV — 1 identical sites and the products of linear chains ofN — i and i — 2 identical sites. The solution for Xt is then JEi = 1
_ y^N-iico)
+ (i - v ) L ^ _ > ) L q y 2 ( c o )
v*WN(oj>) + (1 vfW^co) X{ is different along the perturbed ring due to the presence of the terms depending on i. The properties of Xt can be understood by simplifying eq (5.56). Let 0(iV) be the binding isotherm of a site in a homogeneous ring of N sites. Then )
p.J/;
and eq (5.57) applies to all sites of the ring. The binding isotherm of site i in the absence of site 1, or for Kf = 0, is Ll
Piv-i(ffl)
The function ^f° has the same form as eq (5.44), with ./V replaced by N -1. Hence,
v o w + ( i - v)(i - e ) m ()
(5 59)
()
The site-specific information of Xt is contained in the function X°h which provides a convenient reference state. There are essentially two ways of looking at the perturbed ring. One is to consider it as a homogeneous ring with a perturbed site, while the other is to see it as a heterogeneous linear chain of identical sites (2 through N) perturbed by the presence of site 1 bridging the end sites. The latter interpretation is adopted in what follows. We already know from the properties of the 'unperturbed' linear chain that site-specific effects are dominated by the parameter A, which also sets
5.3 Site-specific effects in Ising networks
277
the relaxation length for the different binding properties of any site relative to that in the middle. If the chain is perturbed as shown in Figure 5.16, what kind of properties are to be expected for the individual sites? How does site 1 affect the other N — 1 sites and the communication among them? If we define the properties of the perturbed chain relative to the site in the middle, like the case dealt with in the foregoing analysis of the unperturbed chain, we obtain the following expression - v)(i - eiN))
(5.60) vd{N) + i - e(N) Except for a scaling term that does not depend on i, XM - Xt is identical to the value in the unperturbed linear chain. The scaling factor does not affect the relaxation length of the chain that retains the same properties. Hence, the presence of site 1 does not affect the way other sites in the linear chain communicate with each other. Quite interestingly, the result embodied by eq (5.60) has a more general validity. Consider the arbitrary perturbation of a linear chain as shown in Figure 5.17. Now an arbitrary Ising network, A, perturbs the linear chain composed of identical sites, from 2 to N, through sites connected to site 1. Again, we seek to find the analytical expression for the binding isotherm Xt of a given site in the perturbed linear chain (/ = 2, 3, . . . N). The definition of Xt demands knowledge of W for the perturbed chain and its contracted form 0*Pf. Using the contracted partition functions we have again W((o) = ^ ( c o ) + ^ ( e ^ a / (5.61) where we recall that co' = K'x, and K' is the binding constant of site 1. The partition function °XF1 refers to a linear chain of N — 1 sites with M
,i —
^M
^ i
—
Figure 5.17 Heterogeneous Ising network representing an arbitrary perturbation of the linear chain through a network A connected to the end sites, 2 and N.
278
5 Site-specific effects in Ising networks
identical binding constants, LW^_l9 times the contracted partition function °^i(A) of network A. The partition function 1W1 is the product of two terms. One is the partition function of a homogeneous ring of N sites, with one site kept in its ligated state, and is given by eq (5.51). The second term is the contracted partition function 1W1(A) of network A. Simple rearrangements yield the partition function of the perturbed chain as follows *?(") = ^KA^Tv^Nico) 0)
+ (1 - -v^W^ico)]
(5.62)
(O
where v= Kl{A)/K, and
= K> ^ W " ) 0 ^ ( )
(5>63)
is the site-specific affinity function of site 1 in network A. The similarity between eqs (5.52) and (5.62) is evident. The site-specific affinity function, wKi{A), replaces the site-specific affinity constant of site 1 and v becomes a function of co. The term °lI/i(A) factors out in the definition of W. The simple case embodied by eq (5.52) is obtained as a special case when the perturbing network A is composed of site 1 alone. The contracted partition function Ox¥ t in the perturbed chain is derived as before from the contraction a))(D'
(5.64)
The partition function ^Wn is the product of three partition functions, i(A)> Lx&N-i and Lx¥i- 2. The partition function 01Wn is the same as that of a linear chain of TV — 1 identical sites, with one site kept in the ligated state, times ^ ( A ) - Hence, the contracted partition function °Wt is given by
Oip
^•(aO = ^ ( a ^ V - V ^ C a j ) + (1 - ^W^co^W^w)]
(5.65)
The solution for Xt is then
Except for the fact that v is now a function of co, eq (5.66) is identical to eq (5.56). The conclusion drawn from the foregoing analysis is that the perturbation does not affect the communication among sites in the linear chain, regardless of the form of network A. In the general case, Xt depends solely on the affinity function of site 1 in network A and the binding properties of the unperturbed linear chain. In fact, carrying over the analysis as in the case dealt with previously we have the expression analogous to eq (5.59)
5.4 The probe theorem _ (1 - "y)(l °vdl
m
as expected.
d(N)) 7
279 (5.67)
(A0
5.4 The probe theorem
We now address the problem of handling perturbations in an Ising network in more general terms. Consider a heterogeneous Ising network, as schematically illustrated in Figure 5.18. Here we make no assumptions about the binding properties of the sites in the network, or about the nearest-neighbor interaction parameters. These site-specific parameters can be assigned arbitrary values and the dimensional embedding of the network is also arbitrary. Consider the section of the network in Figure 5.18 where sites j and /' are connected. If this connection is removed, the network splits into two smaller networks, A and B. Let A be the network containing site / and B the network containing site ;'. Then the following 'probe' theorem holds (Di Cera and Keating, 1994): Theorem: If an Ising network can be split into two independent networks by deleting the contact between two sites, then these sites uniquely specify each other's site-specific properties. If the site-specific binding isotherm of one site is known for the separate
B
ooo
ooo
Figure 5.18 Heterogeneous Ising network that can be split into the independent subnetworks, A and B, by deleting the contact between sites / and /'. The probe theorem states that if the properties of site /' in the 'probe' network B are known when the network is isolated (left), or coupled to network A, then the properties of site j can be derived.
280
5 Site-specific effects in Ising networks
network and the network as a whole, then the same quantity can be derived for the other site. To prove this theorem we consider the affinity function xKj>{AB) for site j ' in the network as a whole AAB1
" K r * The contracted partition function at the denominator is given by °^( A B )(*) = ° ^; (B)(x)W(A)(x)
(5.69)
Ov
where P;(B) is the contracted partition function of the network B isolated with site ; ' unligated, while ^(A) is the partition function of network A isolated. The contracted partition function 1XP;(AB) c a n itself be expressed as a product. Freezing site / ' in its ligated form decouples networks A and B. Network B has a partition function llP7'(B)- The properties of network A are equivalent to those in the absence of B, except for the fact that all configurations with site / in state 1 are to be multiplied by a;y, the interaction constant between sites j and / ' in state 1. Hence, ^ W * ) = lxVm(x)\°VKA)(x) and eq (5.68) becomes 'KnAB)
= *Km)[l
+ lWm(x)off
+ (ajr - l)Xj{A)]
Kjx]
(5.70) (5.71)
which proves the theorem. Measurements of the site-specific properties of site j1 in the isolated network B and in the network as a whole yield information on Xjw and hence -X^AB) that reflects the site-specific properties of site / in the isolated network A and the network as a whole. The interaction parameter o^ can easily be derived in the limit x —> <*> for which Xj(A)->l, since the ratio °°KnAB)/^ Kr{B) is accessible experimentally. If the ratio xfr = xKr{AB)/x Kj>m is defined as a function of x9 then Xm is given by
can
Once Xj(j^ is known, then ^,(AB) and / ' in eq (5.71), so that Y
/(AB)
-
"
1 +
(
j
j
be determined by interchanging /
n
1 + {o]T
which is a useful alternative form. It also follows from the foregoing expressions that ^Q() ~ X() {ojr l)XnB)XKA) (5.74)
5.4 The probe theorem
281
The sign of the difference between the site-specific binding curves of sites j and /' is not affected by the presence of the contact between the sites. If the probability of binding to site j in network A, for a given x9 is higher than that of binding to site / ' in network B, the preferential interaction is still observed when the two sites are coupled in the network as a whole. The significance of the probe theorem stems from the possibility of relating the properties of perturbed and unperturbed networks. Network B can be seen as a perturbation acting on network A, whereby information on the site-specific properties of site / in the unperturbed network A can be derived. More generally, network B functions as a probe for site / in the network A being probed. If the properties of the probe are known when isolated and coupled to A, then the properties of the site in the network being probed can be determined. This is completely independent of the nature of the networks, whether homogeneous or heterogeneous, and is also independent of the particular dimensional embedding. The theorem requires only that the two networks be linearly connected through a single contact. We now derive a rather counterintuitive property of the linear Ising chain, which is a direct consequence of the probe theorem. A linear chain composed of TV distinct sites interacting pairwise in an arbitrary fashion is shown in Figure 5.19. The chain has the remarkable property that its energetics in the site-specific components can be uniquely decoupled if the site-specific binding curve of the end site is known. To prove this, it is sufficient to consider the end site of the linear chain of N sites as the probe of the remaining N — 1 sites. Application of the probe theorem yields X
KN(N) = KN[1 + ((JN-IN ~ 1)^^-1(^-1)] (5.75) The first N — 1 sites of the chain are to be identified with network A in Figure 5.18 and site N is network B. Here xKN^ is the affinity function of site TV in the chain as a whole, XN-i(N-i) is the binding curve of site N — 1 in a chain of N — 1 sites (network A), while 0N-IN is the interKx
K2
Kj
KN
O-O-O-O-O-O-O-O-O-O-O Figure 5.19 Heterogeneous Ising chain composed of N different sites interacting in an arbitrary way. The entire set of 2N - 1 independent site-specific parameters can be resolved from knowledge of the binding properties of site N, or site 1, using the probe theorem.
282
5 Site-specific effects in Ising networks
action parameter between sites N (network B) and N — 1 (from network A). The value of KN gives the affinity function of site TV isolated, which is clearly independent of x and simply reflects the equilibrium binding constant of this site. In the general case, the chain as a whole is characterized by N site-specific binding constants such as KN, and iV - 1 interaction parameters a12, <x23, . . . oN-1N, for the coupling between sites 1 and 2, 2 and 3, . . . N - 1 and N. These 2N - 1 independent parameters define the partition function and can be resolved from eq (5.71), ifxKN(N) is known. In fact, the limits °KN(N) = KN
(5.76a)
=
(5.76b) can be yield KN and oN-1N. When these parameters are known, XN-^N^) constructed for all values of x from eq (5.75). Since .X^-i^-i) is the binding curve of the end site in a chain of N — 1 sites, this site can, in turn, be treated as the probe of a chain of N — 2 sites. The affinity — XN-i^N^)x] and function of this end site is x KN-1(
=
^JV-l[l + (°N-2N-1 "" l)^V-2(N-2)] (5-77) an Again, the limits of KN^N^ yield ^ ^ _ i and ON^2N-I d hence r a XN_2{N-2) f° U values of x. The entire linear chain can be energetically 'sequenced' by iterating this procedure. At each step the site-specific properties of the end site are used to resolve a site-specific equilibrium constant and an interaction parameter. This information allows for construction of the site-specific properties of the end site of a linear chain one unit shorter and so forth. Hence, all Ks and as can be resolved from knowledge of the site-specific properties of the end site of the original chain. This peculiar result only applies to the end site, as follows directly from the probe theorem. Consider the case where X^N) and hence XK^N) is known for a site other than the end site (/ =£ 1, N). Then application of eq (5.71) yields *KN-1(N-1)
X
*Kj(N) = xKj(j)[l + (°jj+i - l)^/+i(;v-7)] (5.78) Here the network from site 1 through j is taken as the probe. The function x Kj(j) describes the properties of site j in a linear chain of length j and is therefore subject to the probe theorem, i.e., *Km
= Kj[l + {ahlj - l)Xhl{hl)]
(5.79)
Hence, *Km
= K,{1 + (<x,y+1 - l)Xj+KN.n][l
+ (ahV
- l)XhKhl)]
(5.80)
5.4 The probe theorem
283
The equation above reduces to eq (5.75) for j = N', or equivalently for 7 = 1. In general, for y=£l, N9 knowledge of XKJ(N) does not provide sufficient information to sequence the entire chain, because the number of variables involved in eq (5.80) exceeds the number of constraints. In fact, the limiting values of x K^N) are °Km = Kj
(5.81a)
-K^^GJ^JOJJ^KJ
(5.81b)
and although the value of Kj can easily be resolved, the interaction constants cannot be decoupled. Only in the case in which all sites interact equally does it make no difference which site-specific binding isotherm is used to resolve the Ks and as. The validity of the probe theorem is not limited to the lattice-gas type of interaction for neighboring sites. If the spin-lattice type of interaction is assumed, with o^ being the interaction constant between sites j and / ' , straightforward calculations lead to KjX] V ( )
/
w
V
/
w K)KJX]
(5.82)
(5.83)
Hence, */'(AB) ~
K
J'(B)-
~ 1 - (1 -
— Ojj,)Xj{A)
(5.84)
and the theorem is proved. Application of the spin-lattice type of interaction to the linear Ising chain gives x
V
_ jf °N-1N + (1 "~ N T
a
N-lN)XN-i(N-i) j (
and again, KN and oN_1N are derived from the asymptotic values of KN(N) thereby allowing a complete energetic sequencing of the chain as in the case of the lattice-gas type of interaction. The probe theorem echoes a well-known principle in statistical thermodynamics whereby information on the state of a system can be obtained from the response of the system to a perturbation (de Groot and Mazur, 1984). The probe represents the perturbation acting 'adiabatically' on a network whose site-specific properties are to be dissected in energetic terms. Information on site-specific properties of the system is derived by measuring the properties of the probe isolated and coupled to the system. It is the perturbation caused by the probe that generates information about the system. The response of the system to the perturbation is measured as the change in the properties of the probe upon its coupling to the system. This suggests that the properties of a specific site can be
X
284
5 Site-specific effects in Ising networks
derived from those of a suitable probe attached to it. The probe theorem suggests a simple strategy for deriving site-specific information from ad hoc perturbations of the system of interest. This provides a connection with arguments discussed in Section 4.5. Extension of this theorem to the case where the probe is attached to two or more sites in the network will provide a key to unraveling the properties of heterogeneous Ising networks in two and higher dimensions. Site-specific thermodynamics promises to be an exciting new approach to the Ising problem in arbitrary dimensions, which still remains one of the toughest unsolved problems of statistical thermodynamics.
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Index
Ackers, G. K. 68, 119, 146, 172, 173, 178, 191, 195, 204-6, 211, 213, 232 Adams, E. Q. 110 affinity function 53, 60, 75, 89, 91, 101, 161, 172, 185 affinity plot 53-5, 62 Ahlstrom,P. 225,228 Akke, M. 225, 228 Allen, D.W. 104 allosteric transition 104 Anderson, C. F. 106, 107 apomyoglobin 238, 239 Arnone, A. 212 Aurora, R. 260 Avnir, D. 46 Ayala, Y. 53, 54, 92, 93, 95 Bai, Y. 246 Baldwin, J. 191,212 Ball, R. C. 46 Bard, Y. 56 barnase 239, 240 Bashford, D. 219 Benazzi, L. 195 Berger, A. 243 Bernoulli number 82 Berry, A. 238 Betz, A. J. 264 Biltonen, R. L. 77, 78, 83 binding capacity 31, 34, 35, 42, 48, 52, 60, 66, 68, 76, 77, 79, 80, 82-4, 96-8, 100, 152, 161, 163, 172, 174, 185 biding curve (average number of ligated sites) 42-5, 48, 54, 60, 61, 63, 64, 68, 71, 79, 80, 89, 97, 100, 146, 148, 161, 162, 172, 173, 185, 268 Blanchard, M. H. 110, 215 Bode, W. 243, 264 Bohr effect 195, 210 Boltzmann distribution 8 Bondon, A. 195 Bose-Einstein statistics 46 Box, G. E. P. 56
Bragg, J. K. 245 Briggs, W. E. 160 Brodin, P. 225 Brown, I. D. 225 Brunauer-Emmett-Teller theory 46 Brunei, C. 195 Bucci, E. 204 Bujalowski, W. 65, 67 Bunn, H. F. 210 calbindin 162-4,166, 224, 226, 227, 229, 230 calcium 162-4, 166, 220, 224, 226, 229. " ^ calmodulin 224 Cantor, C. R. 68 Carlstrom, G. 225 canonical ensemble 8,10, 17, 37 canonical partition function 8, 12, 15, 3, carbon monoxide 61, 65, 67, 105, 196, 19<_ 204,252 Carter, C. W. 212 Carter, P. 238, 241 Cauchy's theorem 80, 81 Caughey, W. S. 136 center of symmetry 73-5,149,150, 213 Chandrasekhar, S. 256 Changeux, J. P. 104, 191 Chazin, W. J. 225 chemical potential 9, 11 Chen, Z.-Q. 78, 82, 83 Chothia, C. 191, 212 Chu, A. H. 68, 210 closed system 3, 10 coefficients a, 80, 86, 87, 147, 148, 154, 156, 160, 172 v, 147, 148, 154, 156, 167 Cohen, S. S. 245 Cohn, E. J. 35 Colombo, M. F. 105 complex plane 80, 85-7, 254 compressibility 31, 33, 34, 36, 76 condensation 43-6 conformational state 1,104 Connelly, P. R. 104
292
Index conservation relationship 119, 146, 152 contracted partition function 123, 129, 130, 137, 165, 177, 188,189, 194, 216, 220, 223, 250-2, 254, 267, 269, 270 contraction domain 250, 253 cooperative binding 59, 61 Cornish, V. W. 238 Crawford, F. H. 32 cytochrome a 136 cytochrome aa3 136, 181 d'A. Heck, H. 51, 58 Dalessio, P. M. 205 D'Ambrosio, C. 195 Dang, Q. D. 54 Daugherty, M. A. 205-7, 209, 210, 257, 259 Defay, R. 32, 42 degeneracy 1,10,12, 48, 252, 254, 256, 257 de Groot, S. R. 6, 36, 283 deHeer,J. 10 De Moivre-Laplace theorem 7, 256 Dennis, S. 264 Di Cera, E. 31, 53, 54, 56, 58, 72, 75, 76, 78, 82, 83, 92, 93, 95, 104, 105,119,121, 123, 130, 135, 139, 146, 149, 186, 191, 200, 202, 203, 213, 214, 225, 233, 245, 246, 249, 253, 258, 263, 279 Dill, K. A. 245 Dirac 6-function 7, 87, 256 direct coupling 218, 233, 235 dismutation 111, 128, 183, 184,192 Dolman, D. 195 Domb, C. 245, 246 double-ratio plot 221-3, 227 double-reciprocal plot 55, 56, 58 Douglas, C. G. 105 Doyle, M. L. 83, 104, 205 Draper, D. E. 238 Drouffe, J.-M. 245, 246, 262 Ebright, R. H. 238 E. coli SSB 65, 67 Edsall, J. T. 35, 110, 215 EF-hand 220, 224 Einstein, A. 191 electron binding curve 136 energy 5, 9, 11, 18, 24, 26, 28, 38, 40, 41 energy level 1, 12, 37 energy state 1, 12 Englander, S. W. 246 ensemble average 4, 5, 15 enthalpy 21,25,48 entropy 9-11,21,23,38,40 equilibrium 28 equilibrium constant 42, 48 ergodic hypothesis 6 Euclidean geometry 37 Euler's relationship 96, 98 Euler's theorem on exact differentials 27 homogeneous functions 24 exact differential 26
293
extensive quantity 11, 17, 18, 24, 25, 26, 28, 29, 32 Farin, D. 46 Feller, W. 7, 37, 78, 256 Fermi, E. 1 Fermi, G. 212 Fermi-Dirac statistics 45 Fersht, A. R. 233, 238-40 Filmer, D. 104 first law of thermodynamics 26, 29, 33, 96 first-order perturbation 220-2, 227 Fleischmann, M. 46 fluctuation 34, 35, 38,135 fluctuation-dissipation relationship 36 Fong, T. M. 95 Forget, B. G. 210 Forsen, S. 225 Fourier transform 257 fractional saturation 50, 51 free domain 250, 253 free energy of coupling 89-92, 111, 112, 200, 201, 203, 204, 206-9, 232, 233, 235, 237, 240, 241,243,244 Freire, E. 83 Fronticelli, C. 204 Gantmacher, F. R. 29, 37, 98, 267 Gaussian distribution 7, 255-7, 261, 263 generalized ensemble 18, 21, 22, 26, 27, 41, 47, 88, 89, 105, 127,139, 247 generalized partition function 21, 35, 57, 60, 75, 80, 88, 91, 97, 99-102, 104, 108, 109, 111, 114, 115, 127,129,139-42, 158,165,169, 182, 191, 202, 210, 215, 220, 223, 254, 267, 269, 270 generalized potential 21, 25, 96,105, 139, 145 generating function 78, 81, 84, 153 Gerard, N. P. 95 Gether, U. 95 Gibbs, J. W. 3, 28, 32, 35 Gibbs-Duhem equation 25, 26, 106 Gibbs' first postulate 4 Gibbs free energy 19-22, 25, 41, 43, 48-50, 58, 88, 99, 127,140, 247 Gibbs-Helmholtz equation 21 Gibbs' second postulate 6 Gill, S. J. 31, 35, 48, 51, 68, 83, 104, 191, 195 Gingrich, D. 205 Glauber, R. J. 245 glutamic acid 215-18 glutathione reductase 242, 243 Gould, R. 248 gradient operator 123, 143,156,157 Gradshtein, S. 82 Gram matrix 37 grand canonical ensemble 13, 17 grand canonical partition function 15, 35 Green, M. S. 245, 246 Hadamard's inequality 98,104 Haldane, J. B. S. 105 Haldane, J. S. 105
294
Index
Hamming, R. W. 249 Hamming distance 249, 250, 256 Hankey, A. 24 Harper, E. T. 260 Hazzard, J. H. 205 heat bath 1,3,5,13 heat capacity 30, 33, 34, 38, 40, 41, 76 Helmholtz free energy 9, 19, 25, 38, 40, 51 hemoglobin 61, 65, 67, 83,105, 191, 198,199, 204, 206-9, 252, 257, 259 Hendler, R. W. 136 Hermans, J. 47,245 Herzfeld,J. 191,245 Hessian matrix 37 heterogeneous network 246, 275, 277, 279, 281 Hill, A. V. 53, 68 Hill, T. L. 3, 7, 45, 46, 51, 89, 202, 217, 245, 266, 267 Hill coefficient 54, 68, 84, 135, 162, 176, 186, 226 Hill plot 53,54,56,68,69 hirudin 92-5, 263 Ho, C. 195,210,214 Hoffman, B. 205 Hofsteenge, J. 264 holomorphic function 80, 81 homogeneous network 246, 258, 266, 268 Hopfner, K.-P. 72 Horovitz, A. 233, 238-40 Huang, R.-R. C. 95 Huang, Y. 206 Huber, R. 264 Hunston, D. L. 119 identical linkage 104 Ikeda-Saito, M. 195 Imai, K. 195 indirect coupling 233, 235 intensive quantity 17, 18, 21, 25, 26, 29, 32 interaction constant 111-13, 128, 158, 170, 182, 191, 192, 196, 210, 216, 229, 230, 266 Inubushi, T. 195 Ising network 245, 246, 248, 260, 265, 279 isolated system 3 Itzykson, C. 245, 246, 262 Jacob, F. 104 Janin,J. 191,212 Jencks, W. P. 233 Jen-Jacobson, L. 238 Jenkins, G. M. 56 Jensen, C. J. 95 Johnson, M. L. 191 Johnson, R. E. 83 Kallenbach, N. R. 238, 246 Karplus,M. 191,212,219 Karshikov, A. 243 Keating, S. 245, 246, 249, 253, 258, 263, 279 Khinchin, A. I. 6 Kirkwood, J. G. 218 Klotz, I. M. 56, 57, 80, 119 KNF model 202, 212
Knowles, J. R. 238 Kojima, N. 136, 181 Koshland, D. E. 104, 191, 202, 212 Kretsinger, R. H. 220 Kronecker delta 256, 257 Kubo, R. 6, 36, 39, 45, 46 Kurpiewski, M. R. 238 A-repressor 172-5, 178 Lagrange's multiplier 7, 14 Langmuir theory 46 Laplace transform 20, 35 lattice-gas interaction 247, 266, 283 Lattman, E. E. 212, 260 law of mass action 42, 43, 47, 48, 70 Le Chatelier's principle 32 Lee, T. D. 87, 247, 254, 266 Lee-Yang theorem 87, 254 Legendre transformation 18 Lesk, A. M. 191 Lesser, D. R. 238 LiCata,V. J. 195,205 Liff, M. I. 238 ligand activity 47, 130, 266 at half saturation 49, 70, 73 ligation intermediate 50, 51, 58, 65, 67,130, 191, 196, 213 Linderstr0m-Lang, K. 35, 218 linkage 30, 88, 90 linkage coefficient 30, 36, 96, 98, 104, 144 Linse, S. 162, 225, 226, 230 Lohman, T. M. 65, 67, 106 Lowdin, P.-O. 130 lysozyme 2 Lyster, R. J. L. 195 Lyu, P. C. 238 IA-P-T ensemble 21 fi-V-T ensemble 16 macroscopic cooperativity 60, 111-13,161, 172, 175, 186 Marky, L. A. 238 Martin, S. R. 162, 225 mass law binding 43, 44, 46 Matthews, B. W. 2, 238 Maxwell relationship 27, 29, 33, 36 Mayer, J. E. 43, 247, 266 Mayer, M. G. 43, 247, 266 Mayne,L. 246 Mazur, P. 3 36, 283 mean ligand activity 70, 72, 73, 79, 103, 146, 214, 253 mechano-thermal ensemble 19, 48, 247 mechano-thermal partition function 20, 22, 41, 47, 49, 50, 57, 88, 99, 110, 127, 140 median ligand activity 70 microcanonical ensemble 10, 17 microcanonical partition function 10, 15 microscopic cooperativity 112, 113, 161 Milne, J. S. 246 Moffat, J. 224,228
Index molecular recognition 238, 242, 245, 258, 260, 265 moments of the B/N distribution 78, 82, 83, 156 ^distribution 153, 154 Monod, J. 104, 191, 195, 202, 212, 235 Montroll, E. W. 245-7, 266 Morimoto, H. 195 Mountcastle, D. B. 83 mutant 162-4, 219, 221, 226, 227, 229-31, 235, 238, 240-4, 260, 263 MWC model 197, 202, 203, 212, 235 Nayal, M. 225 negative cooperativity 60, 65, 68, 83-6, 109, 114, 161,268 Nemethy, G. 104 Neuberger, A. 215, 216 neurokinin-1 receptor 95 Newell, G. F. 245-7, 266 N-P-S ensemble 21 N-P-T ensemble 19 N-V-E ensemble 11 N-V-T ensemble 8 Onsager, L. 247,266 open system 13, 26 Otis, A. B. 195 overall binding constant 59, 76, 88, 100, 102, 110, 127, 128, 140, 158, 170, 182, 191, 197, 210, 222, 226, 229 oxygen 83,105 Padmanabhan, S. 238 pairwise coupling 197, 200-2, 204, 206-9, 218, 230, 233, 235, 237, 240, 241, 243 Palmer, G. 136, 181 Parsegian, V. A. 105 Pauli.W. 35 Pauling, L. 191, 195, 212, 225 Pearson, C. E. 24, 27, 256, 257 Peller, L. 213 Penrose, O. 245 Perham, R. N. 238 Perrella, M. 61, 65, 67, 195-7, 199, 203, 213, 252 Perutz, M. F. 191,212,213 Petrinko, P. 195 Pfaffian form 10, 11, 17-19, 21, 25, 26, 105 Pfeifer, P. 46 Poland, D. 245, 266, 267, 269 polar coordinate 85, 154 positive cooperativity 60, 61, 65, 68, 83-6, 103, 109,114,161,240,268 Pradier, L. 95 preferential binding 106, 107 preferential interaction 107 Premilat, S. 47, 245 pressure 9, 11 Presta, L. G. 260 Prigogine, I. 32, 42 principal minor 29, 30 probability density function IS, 79, 153
295
probe theorem 279, 281, 283 protein folding 231, 235, 260 protein stability 238, 239 Ptashne, M. 172 R state 205,211,235 R2 state 212 Rau, D. C. 105 Record, M. T. Jr 106, 107 reference cycle 88, 90, 99, 108, 109,115 reference system 49, 51, 53, 57, 61-4, 66, 68, 69, 83-6, 98, 103 relaxation length 272-4 response function 9, 11, 20, 26, 28, 36, 60, 89 Ripamonti, M. 195 Robert, C. H. 191 Rogers, P. H. 212 Rose, G. D. 212, 260 Rossi-Bernardi, L. 195, 213 Roughton, F. J. W. 195, 213 Rydel, T. J. 264, 265 Ryzhik, I. M. 82 Scatchard, G. 55, 56 Scatchard plot 55-7 Schechter, I. 243 Schellman, J. A. 51,68, 106 Scheraga, H. A. 245, 266, 267, 269 Schimmel, P. R. 68 Schleich, T. 106 Schottky transition 39, 42 Schultz, P. G. 238 Schrodinger, E. 1 Schwartz, T. W. 95 Scrutton, N. S. 238, 242, 243 second law of thermodynamics 26, 28-30, 32, 33, 76, 96, 98, 135, 144, 152 Senear, D. F. 172, 173, 178 Shaanan, B. 212 Shaw, A. N. 32 Shea, M. A. 119,206 Shibayama, N. 195 Shrager, R. I. 136 Silva, M. M. 212 Simmons, K. C. 212 Simms, H. S. 110 Simonneaux, G. 195 site-specific affinity function 115, 116, 133, 141, 168, 179, 189, 190, 278, 280, 282, 283 site-specific affinity plot 117, 122, 123, 126, 199 site-specific binding capacity 117, 120, 121, 134, 135, 141, 152, 180, 198 site-specific binding constant 111, 127-9, 158, 170, 182, 191,192, 196, 210, 216, 222, 229, 266 site-specific binding curve 115, 116, 118, 119, 121, 130,133, 140, 141, 146-9, 165, 166, 177, 178, 180, 187, 192, 193, 198, 211, 214, 218, 219, 223, 229, 230, 269-71 site-specific cooperativity 108, 116, 134 site-specific Hill coefficient 117, 124, 125, 135, 199 site-specific mean ligand activity 146
296
Index
Skelton, N. J. 224 Smith, F. R. 119, 195, 205, 212, 213, 233 Sodano, P. 195 sodium 53-8, 92-4 spin-lattice interaction 247, 254, 283 Spiros, P. C. 195, 205, 206 Srinivasan, R. 212, 260 stability condition 31 standard transition mode 253, 254 Stanley, H. E. 24, 191, 245, 266, 267 stepwise binding constant 59, 61, 159, 171, 172, 184, 197, 226 stepwise cooperativity 173, 175, 186 Stirling approximation 7 Stockmayer, W. H. 22 Stone, S. R. 264 Strader, C D . 95 Sturgill, T. W. 77, 78, 83 Subba Reddy, K. V. 136 subtilisin 241 super system 3, 13 symmetry condition 73 76, 150, 151 symmetry of the binding curve 12,1'4, 75 symmetry rule 205 Szabo, A. 191,212 Szebenyi, D. M. E. 224, 228 T state 205,211,235 Tanford, C. 106,218 Taylor expansion 81, 82, 153, 257 temperature 1 theorem of residues 80 thermodynamic inequality 33, 34, 98 thermodynamic potential 17, 19, 25 thrombin 53-8, 92-5, 243, 263 Tildesley, D. J. 46 Timasheff, S. 105 time average 4 Tolman, R. C. 6-8, 35, 45, 46
transfer matrix 265-1, 269 transition mode 245,249-53,258-60 transition state 241,244 Tulinsky, A. 264 Turk,D. 243 Turner, B. W. 68, 191 variance of the B/N distribution 84, 85, 87, 88, 157, 163 Bj distribution 154, 155, 168 Viggiano, G. 195,214 Vindigni, A. 54 volume 2, 22, 36, 48 von Hippel, P. H. 106 Wallace, A. 264 water activity 22, 41, 106 Weaver, L. H. 2 Weber, G. 48, 72, 89, 213 Wegscheider, R. 110,215 Wegscheider's principle 215, 216, 218 weighting factor 137, 165, 177, 188, 189, 194 Weinhold, F. 24, 28, 32, 37 Wells, C. M. 53 Wells, J. A. 233, 238, 241 Whitehead, E. P. 72, 174 Wilson, E. B. 7,27,80,256 Wodak, S. J. 191,212 Wu, K. K. 225 Wyman, J. 31, 35, 48, 51, 68, 70, 72, 76, 96, 104, 110, 191, 195 Wyman integral equation 70 Wyman's equal area rule 71, 72 Yang, C. N. 87, 247, 254, 266 Yonetani, T. 195 Zhou, H. X. 246 Zimm, B.H. 245
